The float test is one test but I’ve had perfectly viable starters that didn’t float or float well. To me, a more direct measure of a starters viability is when the starter is bubbly and spongy and when the starter itself rises well within a few hours after feeding. If the starter rises well, so should your bread.
For a starter to rise well after feeding, it can’t be too wet and the amount of flour it’s fed has to be a significant percentage of the amount of starter you’re feeding. A lot more detailed info can be found here:
As for room temp, it plays a role but is not a make it or break it component of creating a viable starter. Yeast just multiplies faster the warmer it is so temperature effects the time it takes to see results and not so much on whether or not you’ll get good results.
I’m using a culture I started in 2013 in September with white flour, but have since converted to whole grain freshly ground hard white wheat (grain purchased from Breadtopia). I keep the culture at 100% hydration, and after feeding, it doubles (or more) in about 5 hours at 72F in a bread proofer (also purchased at Breadtopia) and overnight (8-10 hours) on the countertop, where we keep the winter temperature at about 67-68F during the evening and let it drop to 63 at night. Since we have no central AC, fermentation proceeds more quickly during the warmer summer months, where temperature rises to as high as 80F (and rarely 85F) using old fashioned home cooling techniques (closing windows and blinds during the day, with awnings; opening everything at night). Between uses, I store my culture in the fridge.