Mites are inevitable in fruit fly cultures. Whether they come soon after the first boom or later, after the culture is used, they will get there. I tried my hand at culturing fruit flies and it is a hassle to keep them out. If you order from joshsfrogs.com they have kits with preventative spray. I never tried it but I'm sure it works well. It is fine to feed the cultures out while there are mites in them, grain mites do not harm reptiles and amphibs. I was paranoid about it and only did it once but I fed off a culture that had mites to my colony of bumble bee toads and they were fine. Once you really see the mites though, your culture will completely die within a few days. You want to order more flies now, throw out the culture, and wipe down your surfaces really well and get some mite paper and the mite spray for starting new cultures. Do not start a new culture with an infected culture, they will transfer.
Keeping the flies in a moist and warm environment definitely contributes to them breeding in there faster.