I boil tap water if not filtering to get the residual chlorine out, but I'd been using spring water for make-up. I installed a Pur filter and have been using filtered tap water.
I don't boil water nor heat honey. I use a six-gallon bucket for my primary. I sanitize it, take it to the brew store, load it up with honey, take it home, and add 80°F filtered water up to six gallons. I rehydrate yeast with hot filtered tap water per instructions.
Secondary is a six gallon carboy. I let the mead sit there until clear, testing occasionally. By the time it's clear I've generally tested (and tasted) close enough to a gallons worth. I then rack to a tertiary five gallon carboy for aging and oaking. The mead fills the carboy, so I have no need for adding more liquid.
I haven't sterilized a must yet and, so far, haven't had a problem. All equipment is sanitized before testing or any other operation. I'll be making a cyser this weekend (Schramm's Fall's Bounty recipe), and the potential contamination from bacteria in the apple juice has been a concern. My plan is to focus on healthy yeast to overcome any bacteria, but that's about it.
Lastly, 1 gallon batches are good for experimenting with new stuff, but I like at least 5 to 6 gallons because there is more room to work when aerating, adding nutrients and energizers, and such as that.