I use a half teaspoon (~2g) of sucrose (table sugar) per 500ml bottle for english ales, and 1tsp (~4g) for continental styles (weiss/wit, belgians, hoppy IPAs).
. Even if you cold crash it, there will be yeast to bottle carbonate. You'll only get rid of the yeast if you kill it chemically (Sodium Metabisulphite used for cider etc where you want residual sugar) or filtering your beer. It normally takes a few days to get going, then picks up speed as the yeast wakes up. Give it another week an you'll probably find your beer is fine
I've found that if I leave it in the primary until it is bright (super clear) then it takes longer to bottle carbonate (because there is less yeast in the beer and it's gone more dormant) - normally 2-3 weeks for me. You'll also get less sediment in the bottle
If I bottle it earlier (largely by mistake than by design) when it's still murky/hazy (ie, with more suspended yeast) it can take 1-2 weeks to fully carbonate.
This is in my spare room/garage, where the temperature fluctuates. The temperature at this stage doesn't have a significant impact on the final beer (but does on the speed of carbonation).
Head space looks fine and shouldn't affect carbonation much - beer normally has between 1 and 4 'volumes' of CO2 dissolved in it (meaning for a 500ml bottle, "1 volume" of CO2 will be 500ml of dissolved CO2 whereas "4 volumes" will have 2L of dissolved CO2. The headspace is far smaller than this so has negligible impact on the CO2 dissolving
. Given you have PET bottles, you can give them a squeeze to check on carbonation levels rather than opening