You can do it as long as it will be on a day that you are sure that it will not rain. I would do the paint correction the day before, then the next day paint prep as soon as you get up and apply the coating. That way you are not applying the coating later in the afternoon/evening and have the humidity change create moisture on it that could create stains.
Heat lamps will speed up that process
Yes and no... it depends on the coating, most SiO2 or SiC it will not cure through heat. This is the first thing they tell you when you get certified with OptiCoat Pro. There are a few coatings that need to be applied with IR lamps, and they are mostly sold to Pro installers only. I saw a few recent videos from Jim White and he is using one of those coatings now.
Also, IR lamps big enough for this purpose cost is stupid... Amazon cheap ones are around $200, with decent ones being over $1k.
All coatings flash dry in around 24 hrs, and they shouldn't be touched/washed in the first seven days. OptiCoat recommends applying OptiSeal after 1 hr of application of the coating to act as a sacrificial layer for those seven days. Full cure time take around 30 days and up to 60 days on the cheap stuff out there.