I am so giddy getting to talk about this. I love lore questions. =D
If Tyrael is not human, then why does he look like a human?
When Tyrael became mortal, his body began to form in his fall. He had a measure of control over this process, and as his goals were to ultimately benefit humanity, taking the form of a human was most ideal. After all, he would need to walk among humans and nephalem and fit in. It made sense to blend in.
It's important to note, however, that Tyrael is not a shape-shifter. This is how he looks now.
t's 100% fine that he doesn't identify with heaven, but I don't understand what good he thinks he could do for sanctuary as a mortal. He very well could just be an angel/immortal who doesn't follow the other angels
The issue with this is that the Angiris Council (and indeed, all angels) adhere to their laws so vehemently that there was no room for Tyrael to act outside of them without cutting his ties. The Book of Tyrael touches on this a bit, particularly on how while they have great power in this devotion, it also consistently limits their ability to act.
Also remember that there was once a truce struck between the High Heavens and Hells that neither side would interfere with the matters of mortals. While the Prime Evils have had no qualms about breaking this agreement (as you might expect), the Angiris Council has continued to uphold their end. Heaven's complete and utter devotion to order is both a strength and a flaw. Tyrael realized this, and decided it was best he would no longer be governed by this nature.
They have been battling the TOUGHEST demons for thousands (or more, I don't keep up with the time) of years. He has several millenia of knowledge and is, if nothing else, a formidable foe. Him stripping himself of immortality drastically reduces the overall power of heaven and sanctuary in defense of hell.
Tyrael's strength was, indeed, formidable. But as has been proven in the past, even at his most powerful, he could not take on Diablo himself. There are plenty of powerful beings in Sanctuary (the nephalem heroes being some of them), and the power Tyrael had wasn't serving his purpose. By losing it, he instead gained the ability to help the nephalem directly as an adviser and guide - something he simply couldn't have done by staying in the heavens as an angel.
Think of it a bit like the relationship between Obi-Wan and Luke Skywalker. On the surface, Obi-Wan's death might have seemed meaningless. However, it allowed him to guide Luke in a way he wouldn't have been able to otherwise. There's a parallel there (even if it's not exactly the same).