If "Elohim" can refer to angels and demons, then does Jesus being called "God" really prove anything?

First of all, you need to consider that there were Elohim and Yahweh words for describing God (or divinity in case of Elohim) in the OT.

When you talk about the theos referring to Jesus, you need to understand that it’s a generic word for a divinity (same as John takes logos in John 1:1-16). While Elohists and yahwists (see more on documentary theory) both had the God of Israel in mind, Apostles were using theos both for God of Israel and Jesus of Nazareth.

In other words, Elohim in OT reflects quite the same God as theos in the Apostles’ Koine Greek.

The argument being made against you is based on the "ambiguity" of the word Elohim. As some of the other answers state you can respond by pointing to the context of various passages, particularly those which are used in the New Testament to describe Jesus. However, anyone who is making that argument to deny Jesus is called God will likely respond with their own verse which supports their position.

Therefore a good approach is to agree with them: the word Elohim is ambiguous. In fact "God" is ambiguous in the Old Testament because El also means "God" and occasionally Elohim and El are used together:

God (Elohim) is a righteous judge, and a God (El) who feels indignation every day.
(Psalm 7:11 ESV)

Therefore, to put an end to all uncertainty, Jesus came:

In many parts, and many ways, God of old having spoken to the fathers in the prophets, in these last days did speak to us in a Son, whom He appointed heir of all things, through whom also He did make the ages (Hebrews 1:1-2 YLT)

When God took on human form, He revealed God and He alone is able to reveal God and so He removed any ambiguity or doubt:

No one has ever seen God; the only God, who is at the Father's side, he has made him known.
(John 1:18 ESV)

By doing so you are effectively anchoring your response and the ensuing debate in the New Testament. This is not say you cannot make the same points from the Old Testament. Rather, why debate something which is, at least in the dictionary, correct when you can agree and then shift the focus back to the New Testament.

Great Question! Arguing for the trinity is one of the most complex apologetic tasks you can undertake.

Context is essential to understanding the meaning of any word, especially in Semitic languages (e.g. Hebrew/Arabic) since the same word can have several connected but different meanings. Your questioner seems to be using a faulty analogy since the words are not precisely identical in meaning. Consider challenging with counter questions. Suggestions for greater clarity:

1. Consider focusing your question in terms of specific verses - in context, some are clearly referring to angels and others have always been interpreted by Christian and Jewish interpreters as referring to 'God'. Does your interlocutor genuinely believe that the use of 'Elohim' in Genesis 1:1 is referring to Angels (contra millennia of interpretive tradition)?
2. There is a vast difference between saying that a word can be 'translated' one way and saying it is perfectly equivalent. The Holy God of the bible is not easy to understand and is never fully known (see doctrines of divine simplicity etc.) so the words used in various languages for 'God' are always approximations apart from God's use of his own name. For example, Elohim is plural whereas Theos is simply a generic word for god (not necessarily 'God'). Does your opponent believe that the generic use of 'man' CANNOT we used to refer to humanity (men and women)? If they have not difficulty with that, it shouldn't be difficult to understand how the word for 'strong one' can refer to both angels and God in different verses.

Conclusion: Just as a woman can say she is a member of man-kind to declare that she has the dignity due to all humanity, so Jesus can use Theos/Elohim/God to declare that he is worthy of being worshiped as Creator and Sustain-er of the world.

Some background on the terms themselves: http://embracedbytruth.com/God/Names%20of%20God/Elohim%20and%20Theos.htm

Elohim being used of angels, and elohim meaning "angels" are two very different things. Jesus reminds the Jews that they were called Gods (e.g. Moses), but this referred to judges - and Jesus clearly was using it in the sense of "god, deity," otherwise His response to their accusation is meaningless (the Jews were accusing Him of claiming to be God, as in deity).

The fallacy being committed here is conflating usage with meaning... language 101 failure.