You don't have to carry a balance or really let the cycle close showing a balance to help improve your credit. All that will really matter is how much you are charging in gas per month vs what your limit is. Your usage percentage has an impact, so if you are charging $500 on a $1000 limit for example, 50%, looks high and may make your score a bit lower. So depending on how much your gas is vs your limit, maybe you want to pay full or part before the cycle closes to help bump it up. Usually it's not a huge impact overall though, and also if you had a month close with high usage and the next month close with very little, it corrects itself a bit.
I can only help with 1. You can pay it either before the statements closes or after it closes before the due date. I prefer to pay my credit cards in full after the statement closes. The only time I usually make interim payments is if I have purchased something for someone else and get paid back, I’ll immediate send that money off to the credit card.
I think it depends on the hospital. When I was young and naieve, I spoke with someone in billing who convinced me the best option was to pay it in full with my credit card ‘and just take my time paying it off’. It took me too many bills and too much interest to realize this was TERRIBLE advice. I have never had luck getting my bills reduced, but I have had success setting up payment plans. Now that I am older and wiser 
, I do this for most bills over $1k even though I *could* pay in full with our HSA. If they are going to give me an interest free payment plan, I’ll take that and keep my HSA balance healthy. Most hospitals do have programs for bill reduction/forgiveness if you qualify, so it’s definitely worth a phone call or two!
I'd give your insurance a call. I ran into this a bit with a dentist bill once, they tried to get me for an extra $100... It did have a bit to do with me turning 26 and not being eligible under my dad's healthcare any longer. Basically it was the end of the month that I turned 26, but the dentist verified on the phone in front of me, and I verified and got in writing that my coverage went through the end of the month. But the dentist billed and when I called to talk about the charges, they told me I wasn't covered at all (no insurance because over 26). So I had to go back to the old insurance and sent a message and uploaded dentist bill, written message from them I was covered, original estimate for work, etc. It was a bit of a pain because it was almost 6 months later, so I no longer had an account with that insurance agency to work through, and they had trouble looking me up on the phone. But I sent that through the contact form, got a message back a few days later that I should only owe $$ to the dentist (actually less than the dentist's estimate/what I had paid the day of service, so insurance told me I overpaid) and when I stopped by the dentist office for my next appointment which was around this time anyways, they were like oh you have a credit for $$ (amount insurance told me I overpaid) like magic! Yeah.
Anyways, it can't hurt to verify with insurance if they are done with their part.
I just think it's so stupid that you all have these medical bills to contend withyou shouldn't have to go into debt for your child breaking her arm
if we had opted for the lower monthly cost plan our then employer's offered we would have been on the hook for a large deductible, 20% of all those doctor's appointments, mris, tests.... and $12,000 for the hospital-the equivalent to years and years and years of the higher cost of our insurance. at the same time a friend's child had a serious medical diagnosis. the friend always thought people like us who paid for the higher coverage plans were 'suckers' and he could do better 'banking the difference' (of course it never happened that the difference got banked)-in the first 3 months his kid's expenses were in the hundreds of thousands BEFORE he rolled into a new deductible year. within 6 months they were looking at over $50,000 just as THEIR share of the costs my dd went on to need a second surgery-again low cost to us while his little guy needed several years of treatments to the point of financial destitution of their family.
medical expenses are insane which is why it's so vital if given a choice between insurance plans to weigh out the monthly/yearly (deductibles/max oop costs) potential costs of higher vs. lower premium plans.
dh and i were never so glad to have always opted to pay for higher monthly cost/lower oop cost coverage than when our dd broke her arm about 15 years ago. it was a minor break but it uncovered an issue that necessitated orthopedic surgery. we had an hmo so all we ever paid were $10 copays for each appointment and $25 for the hospital overnight stay (for surgery). it was months of appointments and while i have no idea how much it all came to, the hospital hiccuped on the billing and sent it to us-the negotiated lower amount that the insurance company was paying out was over $60,000
You can always call them and say you need to work out a payment plan. Most hospitals expect this and are happy to work with you. A couple times they just asked me how much I could pay each month and that was that, they noted it and everything was fine as long as I paid what I was supposed to by the date each month we'd agreed on.
