300F is downright cold. I doubt you'd even melt most solder at that temperature.
I usually do 310-325C. Some people like things a little cooler (285-300C) so that they can keep the heat in place longer without damage, while some like things warmer (~350C) so that they can work quicker. Rarely any reason to go over 350C unless you need to solder to enameled wire, in which case 410-425C is about right to melt the enamel off.
Lead free solder melts at a higher temp, so you'd probably need 325-350 to work that at a reasonable pace. Just don't keep the heat on the board too long. The phenolic type material used on monitors especially doesn't like getting too hot. The FR4 stuff used for most game boards is more tolerant. I don't think I've ever damaged a component severely enough for it to not function due to hand soldering heat, though I've broken things due to overheating while desoldering (which tends to take more time).