My 12V DC 30A Power Supply 360W Power Supply is really cheap, and it's worked well for setting up the motors; but now that I'm on to the heated bed, which uses considerably more Ampage than that of just the motors, I'll confess, I'm getting frightened to continue using it; if the summer was a bit longer, maybe it wouldn't bother me, but we're getting into the cold months, and now I'm afraid of ending up using too much ampage just trying to heat the bed in the winter months...(and I don't mean my bed). Is there anything I should look out for in terms of using the either the cheap power supply I already have, or are there certain specs on a new not-so-cheap power supply that I ought to be using instead?
A MK2 heatbed will draw around 12A. The motors and hotend draw only very little power (around 2A, 5A peak), so the 30A supply you have has significant headroom (it is often recommended to derate a power supply by 20%, so a 30A supply would be good for 24A - you're still well under that). It should work fine, even given its dubious provenance.
Winter versus summer should not make a big difference. The largest power draw is during the heat up phase. In winter, the bed will use slightly more power to stay warm, but regardless of whether it is summer or winter the peak power draw during heat up will be the same.
The cheapness of these supplies tends to be reflected in more output ripple (but for heating the bed and running the motors you don't need a very stable voltage) and improper filtering. This may inject noise back into the mains, possibly affecting other equipment nearby. Should this occur, you can just stop using the power supply. However, in my experience, they can deliver the rated power just fine. They're not completely horrible.
Your biggest concern should be whether the wires that lead to your heated bed can handle the current and whether the screw terminals are properly tightened. During the first use, you should check that the power supply does not get extremely hot. If it's so hot it's impossible to touch for more than 1-2 seconds you should not use it.