Why does G‑d hate Edom (Esaw)? Why do you always read negative things about Edom?
The "hatred" of Edom is in reference to the behavior of the Edomites. In the prophetic context, the identity of Edom is bound up in those of its actions.
In Numbers 20:14-21 we see that the Edomites would not allow the Israelites safe passage through its territory, it would not allow it water, and came out threatening Israel by its swords. We see that the Edomites continued to have skirmishes with the Israelites for centuries after (as documented in Samuel, Kings and Chronicles). Eventually they went so far as to partake in Nebuchadnezzar's vicious destruction of Jerusalem and the slaughter of its Jews (Obadiah 1:11-14, Psalms 137:7). Obadiah therefore foretells of recompense for the dastardliness of the Edomites (1:15), and so too do the other Prophets (Isaiah 34:5-8, Jeremiah 49:7-22).
What if I'm descended from Esau, does G‑d hate me? Wouldn't that be unfair?
Do you identify with the House of Edom? Which is to say, do you dignify its ways which are in opposition to that of God? Do you embrace them as your own? Are you set upon the destruction of the Jewish nation and what it represents? If not, in what meaningful way would be a descendant of Edom? It would indeed seem to be unfair if one were indiscriminately cast in with those one is opposed to. I see nothing to suggest that however is the case.
As you can see, it says that none of the house of Esau will survive.
The House of Edom will be destroyed, i.e. its institutions, its ideological/political sway over the world. It does not say that any biological descendant therefrom will be destroyed. On the contrary our Sages have taught that it is only those that perpetuate the ways of Esau to whom the passage applies.
Avodah Zarah 10b:
Once he [Antoninus] asked him [Rebbi]: 'Shall I enter the world to come?' 'Yes!' said Rebbi. 'But,' said Antoninus, 'is it not written, There will be no remnant to the house of Esau?' 'That,' he replied 'applies only to those whose evil deeds are like to those of Esau.' We have learnt likewise: There will be no remnant to the House of Esau, might have been taken to apply to all, therefore Scripture says distinctly — To the House of Esau, so as to make it apply only to those who act as Esau did.
I heard that Edom is Christianity today. But I know many Christians who are very nice people.
The context in which the identification of Christianity with Edom was made is one wherein the Roman Empire and Latin Christianity were inextricably intertwined. As history progressed this entanglement has become somewhat diminished. Insofar as aspects of modern Christiandom or its individuals continue to perpetuate the ways of Esau, the passage in Obadiah would apply. For those that reject the ways of Esau and do not oppose the Jewish nation and what it represents, the passage would be inapplicable.
