if center of pressure moves forward with angle of attack, then does it move forward of center of gravity also at a particular angle?
Longitudinal stability is related to Neutral Point (NP) rather than CP. Exactly because CP moves around, it is not a convenient reference point.

There is no strict requirement for the tail to create downforce for a stable airplane, even though this is a common arrangement.

It may well happen that CP may move ahead of CG and the aircraft remains stable. But it won't keep doing it indefinitely: at some point the wing will stall, CP will move back (this will depend on the wing plan form though), and NP will cease to exist as such. Stability analysis is difficult in such regimes.