When I bought my home the appraisal was one of the items included in the closing costs. I believe this is pretty standard.
For a home inspection, you absolutely should do one even if it's not required. This you will typically pay out of pocket and have to setup yourself (ask your real estate agent if the have any recommendations for quality home inspection professional). You are making most likely the biggest purchase of your life. Get some help to make sure the house doesn't have any major issues.
As far as using Quicken/Rocket, I had started into the process with them as it was my first time ever getting mortgage and they were the familiar name from TV ads. My real estate agent wasn't crazy about it, had heard mixed feedback about them. But I said for now I'll move forward with the pre-approval they gave.
I put an offer on a house, and seller's agent inquired about financing. When she found out it was Quicken she said she had terrible experiences with them, and would be advising seller not to accept an offer with Quicken financing. Said multiple times had situations where Quicken shut down loans last minute causing buyers to scramble for new financing and delaying the closing. That was enough for me to say forget Quicken, and ended up going with different bank (TCF Bank). I don't know if that's a fair assessment of Quicken, but I wasn't willing to take the risk.
My experience with TCF wasn't exactly glowing either, but we got it done. Going forward I'm taking the advice to work with a smaller local lender and/or CU.