A movie and a "cooking" date are small prices to pay for sex. I would do that every day of the week, especially in a situation like he has, when he pretty much has you under his control. So yes, I believe that he would absolutely do stuff like that in order to get sex. And no, you should not be the one initiating and asking him out. It's up to him to prove to you that he's serious, and it's up to you to make him prove that. And by having sex with him right away, you aren't doing that at all. You are an easy mark again and there's absolutely no reason for him to go the extra mile with you because he doesn't have to. I realize why you feel you have to do this, but by giving into that feeling so easily, you are sabotaging yourself from both a) getting him back in a romantic relationship and b) moving on.
The whole "going with your feelings" thing isn't smart. You need to integrate your brain and your heart to make the best possible decisions in your life. Hearts mean well, but often are very very dumb. But overthinking (using too much brain) can be bad as well. Either way, you are making this way too easy for him and you are putting yourself in another bad situation. Make him prove to you he means business and stop having sex with him so easily.