I don't think the bank would agree to reduce your APR. Increasing your limit is much more profitable from their perspective.
If you have a high limit, that creates the possibility that you will borrow more money, which creates the possibility that you won't be able to pay it off, which means you would pay interest in proportion to your credit - this is where the bank makes money, after all. Even if you feel that you can pay it off easily, you can always get hit with a sudden emergency.
For many people, a higher limit also has a psychological effect in that they feel like they have more money "on hand", so they allocate more of their budget to spending.
That said, you should still ask for a lower APR - they might agree due to their own customer retention policy. But it won't necessarily be because your limit went up.
As you say, having more credit increases your score. However, it doesn't increase it that much and credit score doesn't matter that much anyway (I can link questions on here if you're curious). The main effect of the credit score is that if it's really low (as in, a history of delinquency on major loans such as car or house), the bank will either refuse to give you a credit card or demand exorbitant APRs (25%+). But really high scores don't translate into better APR. Mediocre score will get you about 16%, good score will get you 12%. If you have a unique situation (qualifying for a very "high wealth" card like the fancier AmEx cards) then you can get maybe 10%, but I doubt you can get an ordinary credit card with less APR than that. And 10% is still a lot to be borrowing for anything other than emergencies, really. However, by signing up for new cards, you could conceivably get APR 1-2% less than whatever you currently have.
You have a 0% card, which is great! Keep using it as much as you can. It is in your interest as a consumer to borrow as much as possible on the 0% APR (provided you can ensure that you will be able to pay it off) and then move to a different 0% APR card once your current introductory period is over.