What's the justification for believing that God interacts with us today the same way He did thousands of years ago?

What do you mean by "interacts with"? Humans today ARE the same, the rules ARE the same. God judges us the same. But God does not "interact with" us the same, as in, he doesn't speak audibly to us. Your question is really not a question but a liberal rant, and one that is all caused by your inability to differentiate between the rules changing and interaction changing.

Parents may interact differently with their children as they grow, but does that mean their morality changes? Say a parent taught their child that smoking is wrong. As a kid, if they caught the kid smoking they might interact with them one way, and as an adult, a different way. But in both cases the disappointment and disapproval will be the same.

This is an extract from a book written by Donald Knuth:

I have always been taught that God was the same yesterday, today and forever, But then the more I thought about it, the more I realized that God would best be able to communicate sensibly by sending messages that were appropriate for the current time.

For example, we know now that proteins are molecules made up of atoms; but 2000 years ago, people didn't know what molecules were, so Jesus didn't talk about them. Thus it only makes sense that different kinds of revelation are appropriate as the people in the world change.

I believe that there is good justification for believing that God has changed the way in which he interacts with humanity.

In John 9:41, answering those who claimed to be wise:

Jesus said, "If you were blind, you would not be guilty of sin; but now that you claim you can see, your guilt remains.

In Luke 12:

47 “The servant who knows the master’s will and does not get ready or does not do what the master wants will be beaten with many blows. 48 But the one who does not know and does things deserving punishment will be beaten with few blows. From everyone who has been given much, much will be demanded; and from the one who has been entrusted with much, much more will be asked.

Daniel 12:4:

But thou, O Daniel, shut up the words, and seal the book, even to the time of the end: many shall run to and fro, and knowledge shall be increased.

If knowledge increases and we are to be held responsible for what we have, then it follows that the people living in our time will be held more responsible than those living in past times.

Nevertheless, human nature has not changed, and while many superficial aspects of our world are different from ancient times, there are more similarities than differences. As Ecclesiastes says:

“Meaningless! Meaningless!” says the Teacher. “Utterly meaningless! Everything is meaningless.” 3 What do people gain from all their labors at which they toil under the sun? 4 Generations come and generations go, but the earth remains forever.

Eccl 1:10:

Is there anything of which one can say, “Look! This is something new”? It was here already, long ago; it was here before our time.

Ecclesiastes is a direct assault on the idea that human efforts can really change the universe. Like the teacher, we can be caught up in the appearance that the world is changing and accept all kinds of fads. However, in the end, we will find out that nothing has changed.

God, however, has done something new, as Isaiah 43:19 says:

See, I am doing a new thing! Now it springs up; do you not perceive it? I am making a way in the wilderness and streams in the wasteland.

Your question kind of hops around, but I can say that the justification for that belief is the verse you provide plus some others

Hebrews 13:8 ESV Jesus Christ is the same yesterday and today and forever.

Malachi 3:6 ESV For I the Lord do not change

James 1:17 ESV Every good gift and every perfect gift is from above, coming down from the Father of lights with whom there is no variation or shadow due to change.

Yet you, correctly, mention how God's character need not change for his plans to change. But scripture says just as clearly that God's plans have been as unchanging as He has:

• Ephesians 1:9-10 ESV making known to us the mystery of his will, according to his purpose, which he set forth in Christ as a plan for the fullness of time, to unite all things in him, things in heaven and things on earth.

• Acts 17:26,27,31 ESV And he made from one man every nation of mankind to live on all the face of the earth, having determined allotted periods and the boundaries of their dwelling place, that they should seek God, and perhaps feel their way toward him and find him....he has fixed a day on which he will judge the world in righteousness by a man whom he has appointed; and of this he has given assurance to all by raising him from the dead.

• Isaiah 46:8-11 ESV Remember this and stand firm, recall it to mind, you transgressors, remember the former things of old; for I am God, and there is no other; I am God, and there is none like me, declaring the end from the beginning and from ancient times things not yet done, saying, ‘My counsel shall stand, and I will accomplish all my purpose,’ calling a bird of prey from the east, the man of my counsel from a far country. I have spoken, and I will bring it to pass; I have purposed, and I will do it.

If then, your question goes further in asking "His nature and eternal plan are unchanging, but is he doing the same stuff now as he was then?" Then I think that's a bit too broad. I hear all the time about how Muslims in the middle east are seeing Jesus in their dreams. I still hear about miracles and I still hear about how God is speaking believers everywhere, but that is all anecdotal.

This reference to the book of Joel that Peter gives in one of his sermons seems to say that God will still be doing his cool, miracle-ish type stuff all the way to the end of time.

Acts 2:17 ESV And in the last days it shall be, God declares, that I will pour out my Spirit on all flesh, and your sons and your daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams;

What's the justification for believing that God interacts with us today 
the same way He did thousands of years ago?

In a word: Experience.

Consider the value the following men place on experience:

“This grace of God is a very great, strong, mighty and active thing. It does not lie asleep in the soul. Grace hears, leads, drives, draws, changes, works all in man, and lets itself be distinctly felt and experienced. It is hidden, but its works are evident.” ― Martin Luther

“The gospel is not a doctrine of the tongue, but of life. It cannot be grasped by reason and memory only, but it is fully understood when it possesses the whole soul and penetrates to the inner recesses of the heart.” ― John Calvin, Golden Booklet of the True Christian Life

“And Festus said with a loud voice, Paul, thou art beside thyself.” Acts 26:24. And so say all the world, the men who know not God, of all that are of Paul’s religion: of every one who is so a follower of him as he was of Christ. It is true, there is a sort of religion, nay, and it is called Christianity too, which may be practised without any such Imputation, which is generally allowed to be consistent with common sense, —that is, a religion of form, a round of outward duties, performed in a decent, regular manner. You may add orthodoxy thereto, a system of right opinions, yea, and some quantity of heathen morality; and yet not many will pronounce, that “much religion hath made you mad.” But if you aim at the religion of the heart, if you talk of “righteousness, and peace, and joy in the Holy Ghost,” then it will not be long before your sentence is passed, “Thou art beside thyself.” ― John Wesley, Sermon 37 "The Nature of Enthusiasm"

In different ways, they are arguing against a 'dead' orthodoxy that merely consists of intellectually assenting to propositions to be 'believed'; but rather for an experiential faith that makes a profound difference to the life of a believer. As these men all believed in sola scriptura, what constitutes 'valid' experience is of course determined by scripture (i.e. 'how God dealt with the ancients') - this necessarily includes 'how God relates to us today'.

The preceding is the substance of my answer, but for the sake of further clarity: From the sermon quoted earlier, Wesley goes on to say, in warning on the dangers of 'enthusiasts' receiving revelation by...

...what is justly called an extraordinary manner: I mean, by visions or dreams, by strong impressions or sudden impulses on the mind. I do not deny, that God has, of old times, manifested His will in this manner; or, that He can do so now: nay, I believe He does, in some very rare instances. But how frequently do men mistake herein! (Ibid., emphasis added)

Wesley at least, affirms that at least some of the 'supernatural' ways God has related to men in the past do still happen, but are not necessarily the norm, rather:

Perhaps some may ask, “Ought we not then to inquire what is the will of God in all things? And ought not His will to be the rule of our practice?” Unquestionably it ought. But how is a sober Christian to make this inquiry? to know what is the will of God? Not by waiting for supernatural dreams; not by expecting God to reveal it in visions; not by looking for any particular impressions or sudden impulses on his mind: no; but by consulting the oracles of God. “To the law and to the testimony!” This is the general method of knowing what is “the holy and acceptable will of God.” (Ibid.)

So this is the norm (as per the model of the Bereans - cf. Acts 17:11), however just because 'visions...dreams...strong impressions or sudden impulses on the mind' from God are 'very rare', they are not non-existant.

It is actually a reasonably common mistake, upon cursorily reading the Bible, to assume that the way God normally interacts with man is overtly miraculous. Such a view (a seemingly implied premise of your question) focuses on the 'spectacular portions' of scripture ignoring the fact that over the historical period covered by scripture, the miracles, visions and dreams etc. presented are actually few and far between (the periods of the Exodus and the ministry years of Jesus being the exceptions that actually prove the rule). Scripture is not just a 'highlights package' of God's 'greatest hits', but it does actually record (significantly extensive) periods where nothing much is happening on the miracle front - the people of God are chronicled as proceeding through life in a fashion that is either characteristically obedient or disobedient to earlier revelation, but not necessarily 'encountering' God until (or sometimes, not even when) some crisis eventuates. Prophets receiving their prophetic revelation may be viewed as a significant exception to this, but for the vast majority of people contemporaneous with the prophets, they could easily have viewed the prophets and what they said as just a guy mouthing words unless there was some miraculous confirmation or direct fulfillment of foretold events (which often didn't happen for generations after the fact) - indeed at least in Jeremiah's case, that is exactly what happened.

In regard to some of the other issues you raise -

Many societies today are vastly different from the societies of the ancients.
Take the rise of democracy, for instance, or the rise of various economic systems 
like capitalism and communism. Take the advent of Science and how
electronic/computer technology has become so pervasive and significant. The 
controversies surrounding abortion and contraception absolutely could not have 
happened two thousand years ago.

The Bible itself portrays:

• People groups that undergo significant cultural transformations - including, but not limited to, the development of various technologies and operating under different political and economic systems - but God's interactions with those people groups not changing significantly as a consequence of these particular changes, but rather -
• God changing the way he relates to people according to how they relate to Him. In the book of Judges for example, viewed at one level, God changes the way He relates to His people from chapter to chapter - alternately allowing them to be oppressed and then delivered (of course, viewed more wholistically, God is completely consistent - it is the people who He deals with that are inconsistent). The Amorites (cf. Genesis 15:6 & book of Joshua) and Ninevites (book of Jonah) are further salutory examples of this.
• Significant and lasting changes in the way He relates to people according to 'Covenants' he makes with them (particularly Adam, Noah, Abraham, Jacob/Israel, Moses & the children of Israel, David and of course the New Covenant offered through Christ to all peoples of the Earth).

Summary: the Bible itself portrays a variety of ways that 'God behaves towards mankind'. Theologians who affirm sola scriptura necessarily bind themselves to scriptural (especially New Covenantal) models of this interaction - for them to deny that God would 'interact with us today the same way He did thousands of years ago' would be inherently inconsistent.