Are our consciences trustworthy?

Yeah, our consciences are trustworthy. Our conscience will always tell us the truth. It is our passions which cloud our ability to follow our conscience. Original sin is that which clouds our conscience, makes us follow our passions (like answering questions on StackExchange when we should be working).

I asked my religious ed. class (7th and 8th graders) last year, "what is your conscience?" lots of them say "it's the little angel and devil who sit on your shoulders and tell you what to do." But no, that's not your conscience, your conscience is:

"Deep within his conscience man discovers a law which he has not laid upon himself but which he must obey. Its voice, ever calling him to love and to do what is good and to avoid evil, sounds in his heart at the right moment. . . . For man has in his heart a law inscribed by God. . . . His conscience is man's most secret core and his sanctuary. There he is alone with God whose voice echoes in his depths."

CCC 1776

Total Depravity, however, is a concept that runs contrary to Catholic doctrine so I can't comment on it as a proof of anything and well if you manage to test everything and find that you still have something (1 Thessalonians 5:21) then you couldn't be totally depraved, eh?

Total Depravity idea hearkens back to those old heresies that say one thing is bad and another thing is good. Original sin just says, we started out good and now we're bad but at least we have something to aspire to. Which is why the doctrine of the Immaculate Conception matters. Mary is a type of the Church as it should be, the spotless bride of Christ, in imitation of her, we become more perfect disciples. If she was free from original sin, conceived without sin, it was because it was possible for a human body to be sinless.

We on the other hand, have got the yoke of sin on our shoulders and our consciences will lead us to cast it off and to strive for sinlessness.

In short:

No

Do not trust your conscience.

Regarding this verse above, our hearts and our consciences are the same thing, biblically speaking. They are part of our "soul" (in biblical terms). (See this question for distinction between the soul and the spirit.)

To show that these are the same, I'll go back to the original Greek, so that we can understand the original intention.

Conscience defined

If we look at the Romans 2:14 (from the question) and examine the Greek, we find the word for "conscience" is syneidesis. However, the definition is a bit foggy:

1) the consciousness of anything
2) the soul as distinguishing between what is morally good and bad, prompting to do the former and shun the latter, commending one, condemning the other
   a) the conscience

Here, we see that the word can mean either our "consciousness" and our "conscience" (our sense of right and wrong). In context, we can determine that it is the second definition that Paul is intending to use. He's comparing this to a man's heart, so we can safely translate this word as conscience, meaning the second definition.

Heart defined

Romans 2:14 (from the question) uses the word "heart" (as in the laws are written on our hearts). The Greek word for this is kardia

This word clearly references the physical organ. However, it also has a bit of a metaphorical use (which is comparable to its use in English). Ignoring the "organ" definitions, we see:

1) the vigour and sense of physical life
2) the centre and seat of spiritual life
   a) the soul or mind, as it is the fountain and seat of the thoughts, passions, desires, appetites, affections, purposes, endeavours
   b) of the understanding, the faculty and seat of the intelligence
   c) of the will and character
   d) of the soul so far as it is affected and stirred in a bad way or good, or of the soul as the seat of the sensibilities, affections, emotions, desires, appetites, passions

When Paul uses the words "heart", he seems to be talking about the second definition: "the center and seat of spiritual life". More specifically, he's referring to the soul as it's affected by morals, emotions, desires, etc.

Conscience = Heart

With these two words defined, we can clearly see that he's using two separate words (one literally and one metaphorically) to illustrate a concept: God's laws are inside of us, written on our hearts, implanted in our conscience. (Note the three "places" in that sentence are really the same concept.)

In the same sense as above...

Do Not Trust Your conscience!

The verse you show in your question is a great example of people who do not have the law. In context of the scripture, this means that they are not Jews. The Law (Old Testament rules and regulations) were not given to the Gentiles, so they do not have any guiding standard to determine right from wrong.

Instead, God has given them their conscience as a guide. He has written the laws on their hearts--implanted them in their consciences. Does this mean that their conscience are infallable? Absolutely not!

While they have the laws written on their hearts, in their humanity, they are fallible (All have sinned, etc.). However, these non-Jews / non-Christians who do not have the law are subject to their consciences instead.

Romans 2:15b (NASB)
...their conscience bearing witness and their thoughts alternately accusing or else defending them

This does not mean that they will be free from sin. However, if they earnestly sought to do what is right in God's eyes, they will be justified even though they did not know what was "right" or "wrong" (per the Law).

Does this mean we can trust our conscience? I would say a huge, incredible No!! The reason is because we have been given explicit guidance on what is right and wrong: specifically, the Bible. The Gentiles in your verse did not know right from wrong, but we do.

That's why we're commanded to test the spirits:

1 John 4:1a (NIV)
Dear friends, do not believe every spirit, but test the spirits to see whether they are from God

Like the Heart/Conscience metaphor above, here Paul tells us to test the "spirits" to determine whether they are from God. We are to determine whether the internal pullings and tugs we hear inside of us (either as self-talk or as emotional affects) are in accordance with God's will. How do we determine God's will? By reading the bible.

If you have a "spirit" telling you to do something contradictory to the Bible, that thought or emotional tug is not from God.

Our hearts are deceitful

One final verse I'll throw in here for all the naysayers:

Jeremiah 17:9 (NIV)
The heart is deceitful above all things and beyond cure. Who can understand it?

Do NOT trust your heart.

Summary

1 John 4:1a (NIV)
Dear friends, do not believe every spirit, but test the spirits to see whether they are from God

The idea with this verse is that we should question what goes on inside of us. If we hear something that is telling us that we should go one way or do a given deed, we need to test that guidance to determine whether or not it is from God.

Yes

The word conscience comes from the Latin conscire, a compound of con (with) and scire (to know), meaning “to know together with,” “joint knowledge with another.”

Your conscience is your ability to know right and wrong in connection with the laws of God that have been written upon your heart at your new birth of the Holy Spirit. Heb. 8:10-11

Your conscience is the result of God's breath and life within you.

Genesis 2:7 (NIV)

7 Then the LORD God formed a man[a] from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being.

Job 32:8 (NIV)

8 But it is the spirit in a person, the breath of the Almighty, that gives them understanding.

Proverbs 20:27 (NIV)

27 The human spirit is the lamp of the LORD that sheds light on one’s inmost being.

The prophets relied on their consciences

Romans 2:14-15 (NIV)

14 (Indeed, when Gentiles, who do not have the law, do by nature things required by the law, they are a law for themselves, even though they do not have the law. 15 They show that the requirements of the law are written on their hearts, their consciences also bearing witness, and their thoughts sometimes accusing them and at other times even defending them.)

Romans 9:1 NIV

I speak the truth in Christ—I am not lying, my conscience confirms it through the Holy Spirit—

2 Corinthians 1:12

Now this is our boast: Our conscience testifies that we have conducted ourselves in the world, and especially in our relations with you, with integrity and godly sincerity. We have done so, relying not on worldly wisdom but on God’s grace.

Weak consciences

Your conscience can become weak and defiled if you do not keep it clean.

1 Corinthians 8:7

But not everyone possesses this knowledge. Some people are still so accustomed to idols that when they eat sacrificial food they think of it as having been sacrificed to a god, and since their conscience is weak, it is defiled.

Titus 1:15

To the pure, all things are pure, but to those who are corrupted and do not believe, nothing is pure. In fact, both their minds and consciences are corrupted.

Here's a great article if you want to get into some really deep reading regarding your conscience.