First, do not "reflow" this card, assuming that by "reflow" you mean put it in a domestic oven. That's a good way to turn a working card into a dead one. The worst case scenario here is that you graft some case fans onto this thing.
What I...
The [H]ard way, obviously.
The cleanest way to deal with this would be to desolder the connector. It'd be pretty easy to do with a hot air station, but you could probably use a regular soldering iron and coax it off of there. Once you hand the...
This sounds to me like a BGA failure, or damage to the board.
It's also possible that it's something like a cracked solder joint on some other component, but if it's intermittent, or it works if you press on certain components, it's probably...