Hi and welcome to the forums!
My philosophy is that if you wouldn't buy it without a credit card, then you shouldn't buy it just because you can put it on a card. I would also be nervous about putting $10k on a credit card and carrying a balance, even at 0% interest, unless I had the $10k sitting in the bank to pay it off.
I'm not sure from your post whether you do already have $10k saved up to spend on home improvements, and your $1k/month rule is just on how you spend it. If you don't have the money saved up, I definitely would not spend on credit without the cash to back it up, even at 0% interest. Or maybe you do have the money and are deciding whether to spend it on home improvements or paying off your mortgage faster. (Actually, paying off your student loans first is probably the best idea, since the interest rates are likely higher, and student loan debt can't be discharged in bankruptcy.)
Something to think about... You want to spend ~$10k on home improvements on a house that you are very likely to sell in 3-5 years, and you seem unsure whether you'll get that money back in the sale price. I know you want to enjoy the improvements for 3-5 years, but think about when you buy that new home, will it also have those upgrades (which means they'll be factored into the price), or will you also want to do them for your new home? Presumably you'll live in your new home for more than 3-5 years. I would just think about this. Maybe it is worth it to you to spend this money now, or maybe it isn't.
If you were determined to do this and you had the cash in hand, I would just do it all at once and pay it off the same month. If you need to save up the cash, I would do each piece after I had saved up the money. However, this would take longer, which would mean you have less time to enjoy the improvements. . . . So again I'm not sure if you should even spend the money on this current home.
I can easily imagine spending $3-4K in one monthy, including a new stove, painting my living/dining room, and maybe just my kitchen and bathroom floors. I could immediately pay it off, but I'd rather spread it over 2-3 months, while I'm not being charged interest.
I guess I don't see the point in spreading it out over 2-3 months if you can pay it off immediately. The max interest you can make in an online savings account is ~1% (annual rate), I think. Wouldn't be worth it to me.
I'm not sure if you already understand this distinction, but utilization is calculated based on the statement balance reported. So say you were planning on putting $10k on the card and paying it off the same month. You could pay off the entire amount (or a partial amount) before the statement cutoff date, and your utilization is based off the statement balance. But I think you are probably thinking of carrying a balance from month to month.
If you do decide to do it all at once and carry a balance on your card, your utilization will shoot up, and your scores will decrease. So you should apply for new cards before doing that.
As for optimal number of credit cards, I've heard 3 is a good minimum. See #1 here. A lot of people on these boards have way more than 3 cards, which is fine if you can manage them. Personally, I have 3 cards and I'm not planning on getting more, because I don't want to maintain even more accounts.
I would look at your spending and determine which categories you spend the most in, and then look for cards that have rewards in those areas. Do you spend a lot on groceries, gas, restaurants, Amazon, travel, etc.? Once you figure out your spending, we can help you pick a card to maximize rewards. Sign-up bonuses should also be kept in mind. You could use your home improvement spending to meet spending requirements. (But again, that should not be a reason to spend money if you weren't going to do so already.)