I had the impression that 何 is pretty common kanji and that it is used in situations as above (after all, 今 is there, so it is not that it wanted to be hiragana-only). Do I understand it right that it could be written as 今何してる and if yes, what is the reason it is written 今なにしてる instead?
As ジョン said, 今なにしてる could be written as 今何してる. However, I can think of two plausible reasons why they write 今なにしてる instead of 今何してる.
First, hiragana gives more informal and casual impression than kanji. I do not know the overall tone of text used on Facebook, but I assume that it is quite informal, judging from the colloquial expression してる (instead of している), so writing なに in hiragana is aligned with this casual atmosphere they want to give.
Second, some people prefer to avoid writing two kanji letters in a row when they do not form one word, because two kanji letters suggest that they form a Sino-Japanese word. Compare 今何してる to 勉強してる and notice that 今何 may look like one word, especially if you are not paying attention.
Other answers are correct, I just want to add one more thing:
My gut feeling is that なに is sometimes likely to be written in hiragana because it's a simple interrogative word. Most (simple) interrogative words, like どれ, なぜ, どう, いつ, どこ either do not have kanji renditions or their kanji renditions are rarely used. The same can be said for other simple determiners, like これ, この etc.
だれ does have a commonly used kanji rendition, but I claim that for the same reason as なに, it's sometimes likely to be written in hiragana.