So 15A cables, etc. are OK?
Yes, 15A is adequate. I'm an EE, I borrowed a friend's Kill-A-Watt meter overnight once and the current peaked at just over 13A. That's what I expected to see; a standard wall outlet is fused (or, more often, circuit breaker-ed) at 15A, and they want to leave a LITTLE bit of headroom in case you've got a lamp or something else plugged into the same circuit. Nobody likes nuisance tripping.
Cheap power strips use cheap devices; a cheap circuit breaker or MOV (Metal Oxide Varistor, the device that provides surge suppression) might work fine when new, but their performance (especially the MOV's performance) can degrade pretty badly over time.
Get yourself a new one, or just a new 15A extension cord, and you'll be fine. 14-gauge wire is fine unless you're really concerned about keeping the extension cord very cool.