1) He is not picking up the phone. (Can I use it for mobile phones as well?) - that's out of date.
There's no answer/ He's not answering/Not in. etc. (Others will give you more examples)
2) He is not taking my call.
Yes - if he is declining/refusing to speak to you.
3) He is not picking up my call. The situation in which you might say this is so complicated and rare...
4) He is not accepting my call.
Another way of saying someone doesn't want to speak to you, period; or because it is a reverse charge call and he declines to bear the cost of the call.