I read the comments and might not have much to add in wisdom, but here's been my experience, in relation to your questions:
1. Being our first home, how much should we have saved for unexpected repairs?
- As mentioned, it is hard to say. I got a good inspection, and he said "you'll probably need a roof in about 5 years" and pretty much exactly that much later, I needed on. ~$5k. He didn't know that the heat pump would go... but since it was ~17 years old, it went. They can last a lot longer than this, but I wasn't so lucky. Age of the home and age of these items may help you to guess how long they'll last. My water heater also went... may have been the original. So it makes sense that it'd die after this much time.
2. What is a reasonable expectation for closing costs? Since it's a sellers market, should we assume that the sellers will not pay closing costs?
- As others have said... for my $156k home I "paid" $7500 in closing costs, but about half went into escrow for prepaid taxes/insurance. Looking at the comments below, maybe I paid too much? I do know mine included $500 which was a one-time fee that let me not have PMI payments despite a 5% down payment. 1% title fee was $1600. I'd have to dig to see what else I ended up paying for...
3. Should we extend our lease month-to-month past February to save more money?
- Broken record here, as others have said, buy the house at the right time for you. That is to say, when you found a house that you're sure you want to stay anchored to for the next ~7+ years. You wouldn't change cars if it meant you'd lose $5-15k and end up with the same car in a different place. With a home, this is a reality! You pay a lot up front, and more interest up front. For me, it was a mixed bag. I've been renting out bedrooms for income, but also had higher than expected repair costs. I'm past the 7 year mark, and I'm starting to feel OK about the house financially... compared to renting. But up until this point, it was the more expensive option!
I hope my anecdotes add a touch of humanity to your wisdom pool!