Yes, it's normal. At some point soon, you'll learn to out the worry to one side, accept it'll always be there but know it's not helping you or her.
Your post is exactly why being a second time mum is so great. You just don't care about that stuff!
Worried I'm not doing things right - very normal, going back to work is the best thing for me as at least I know that's something I can do. There's an outcome I can see and if I work hard and use my knowledge I get results. Babies - Ha! Follow the rules and work as hard as you like, you'll still have days they can't be put down, won't feed and think sleep is for wimps...
Worried about what I should be doing - fed, warm, loved. Job done.
Feel like I don't know anything = motherhood
Number of naps - all babies are different. Dd is 11m and managed 10mins this am and 30 mins this afternoon. 
Watch for sleepy signs and go with the flow.
Fall asleep on their own - IME you can't teach this. They all reach it at a different time, just like any other milestone. Dd still needs me sat next to her to fall asleep but will sleep through about 50% of the time.
Playing - watching you potter about and singing or chatting to her is fine. She may just be starting to bother with toys.
Tummy time - mine hated this and so I rarely bothered. Still crawled/walked early.
Sleeping - the no cry sleep solution is a good read but again, some sleep, some don't. Stop thinking of it as something you have to fix. DS woke every 45 mins for 5 months. There wasn't anything I didn't try. Slept through from 8m after I'd given up.
Teething - calpol, Ashton and parsons, cold teethers.
The panicky feelings are the only worrying thing in your post. No one knows how to parent. We all just make it up from books/advice/our own childhoods or trying to do the opposite. Don't try to be super mum. Good enough is more than fine.
Just try to enjoy being a mum as much as you can. The rest will fall into place in its own time.