Does Exodus 3:14 explain who God is to everyone regardless of time and culture?

God is referred to by a number of names in the Old Testament, including Elohim, El Shaddai, Yahweh and, in Exodus 3:14 and Hosea 1:9, Ehyeh.

The Book of Exodus includes the following dialogue between Moses and the God of Israel:

Exodus 3:13-14, NRSV: But Moses said to God [Elohim], ‘If I come to the Israelites and say to them, The God [Elohim] of your ancestors has sent me to you, and they ask me, ‘What is his name?’ what shall I say to them?’ God [Elohim] said to Moses, ‘I am [Ehyeh] who I am’. He said further: ‘Thus you shall say to the Israelites: I am [Ehyeh] has sent me to you’.

Notice here that the narrator refers to God by the ancient name Elohim, but God refers to himself as Ehyeh, which is best translated from Hebrew as 'I WILL BE', although more commonly (as shown in the above NRSV Bible extract) as 'I AM'. Commentators are still coming up with explanations for the meaning of this obscure name, which appears to be derived from the Hebrew verb 'to be'. Whether read in English as 'I AM', and even less when read as 'I WILL BE', this name does not really explain who God is.

John's Gospel, written in Greek, frequently uses plays on words, including having Jesus use the words ego eimi ('I am'). Greek readers understood ego eimi as a synonym for the Hebrew God, so John 18:5-6 has Jesus say "I am", and they fell back as if Jesus had intentionally told them he is God. (Some English Bibles translate this as "I am he," but the word 'he' is not in the original Greek.)

By this, 'I AM', God declared that He is self-existent and eternal. He had 'no beginning', He just was who He was.

God used this name, it seems, partly because the world was fully flooded by idolatry at that time and the Jews were being singled out as the only people to preserve the truth about the God who is 'One'. Under Moses God showed himself to be more powerful than the gods of the great Egyptian power. This would to all who worshipped other gods that they can't stand in conflict with the One God.

The Gentiles had no excuse in rejecting and suppressing the irrepressible truth of this manifest eternity and power of God, for:

18 For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth. 19 For what can be known about God is plain to them, because God has shown it to them. 20 For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse. 21 For although they knew God, they did not honor him as God or give thanks to him, but they became futile in their thinking, and their foolish hearts were darkened. 22 Claiming to be wise, they became fools. (Romans 1:18-22)

This answers the first question, 'I AM' is the first name by which all mankind always first recognizes God.  Every person, even as a child, while looking into the sky, or sea, has learned this name. Of old they turned those views of sky, sea, sun etc., into an idol. Today we suppress this truth in other ways, not necessarily in an outward idol, though some still do that.

For the second verse about the new covenant written in the heart, this has to do with new birth by faith in Christ. Although Many saints in the Old Testament had faith in Messiah and therefore had a new birth ahead of their time, the laws of Moses were written on stone to signify that they were not yet written by faith into the hearts of the listeners. This is why God was partly unapproachable under those shadows of the Old covenant. In the New covenant those laws have become, principles of life, eternal life springing up from the soul, written onto our heart through a new birth by the Spirit. 

7 Now if the ministry that brought death, which was engraved in letters on stone, came with glory, so that the Israelites could not look steadily at the face of Moses because of its glory, transitory though it was, 8 will not the ministry of the Spirit be even more glorious? (2 Corinthians 3:7-8)

For you have been born again, not of perishable seed, but of imperishable, through the living and enduring word of God. (1 Peter 1:23)

This is how words that were only written on a sinners conscience before, words that killed and condemned, now are words of life in a new nature.

To answer the second question, this is even more relevant today as under the New Covenant the promise of Messiah to Abraham becomes a blessing to the whole world. (Gen 12:2)

For more descriptions of what God is like see this post.