Can heat exhaustion get worse or develop into heat stroke after going in a cool environment?

Yes, it can on occasion if the body doesn't cool fast enough. This is why the first-line treatment for heat stroke in a fully conscious patient is to administer cold fluids orally, and to administer cold fluids IV if the patient is not fully conscious. (though one should never administer anything orally to an unconscious individual). This is because the cold fluid works to lower the core temperature much more quickly than the surrounding air is able to lower body temperature.

No, if you go to cool down, it doesnt progress like that

it depends on the situation, theyre all different

Stay inside you freaking fâg