Hello. I think your examples are red flags and yet they can be worked on if your partner is willing. But I will tell you your partner doesn't care about you because your partner doesn't act like it. Since I grew up with abuse, I was really clueless and put up with "micro" abuses and dismissed them as "I'm so self-centered and need to be better and just give in." This just led to worse and worse and worsening intimate partner abuse.
Pushing for sex when you don't want it, and then continuing to push is not caring about you at all. I once had the best lover and what made him stand out is he would "check in" with me during sex; every little move was an invitation and he waited for the yes. Sometimes I didn't respond with a yes, and he stopped, even when he was sexually frustrated. He managed his own desires, because the sex was an experience to enjoy. Having sex with someone who really doesn't want to have sex with you just feels so --- ick, gross, who would want that? My best lover was caring toward me by always checking in if I was ok and if I responded to his next move. Needless to say-the sex was pretty hot and passionate when it happened. Who wouldn't sexually blossom with that kind of care and understanding. Being pushed for sex in the way you describe feels more like "porn sex" driven rather than intimacy driven. Porn is kind of fast and hard and just leaves you feeling like a thing used, like someone just went to the bathroom on you.
One thing to ask is when you say no, and he doesn't push for more, what does he do? Does he act depressed, or like you don't care about him, or get angry? These are big flags. You should not be made to feel guilty for not being in the mood, and certainly anger would be a big red flag.
Your partner sounds very young like a teenager. Not wanting to pay attention to you, but demanding all attention be on them. If they are unwilling to listen to your cares and concerns and make adjustments, then I would suggest relationship counseling. Your partner may have some deep issues that keep them regressed in the teen boy stage. If they don't want to work on it, then you have a decision to make.
For what it is worth, those are my thoughts! Good Luck!