1. Raiise, the value of top pair with AK goes down so much in limped pots.
2. Raaise, how is this a question?
3. Is possible to check in a very specific tables. You should be able to articulate to yourself why you are exactly checking and how you can play it better in a limped pot vs. a raised pot. If you can't do this (it should be about 80 % of the time, typically requires a couple of guys who are willing to put money in two streets with TPNK in limped pots), raaise!
4. Call is a decent option here. You have the position now and post shouldn't be too hard to play.
The fact that you asked about the first two scenarios makes
tournament players.