If you stick with your starting house + professor, statistically speaking 2-3 of your units will be endgame liabilities, just from RNG stat progression, not even counting any dumb choices you make like not having a tank or whatever. (Edelgard makes a fine tank if you do nothing and just let her level, fyi)
If you look at what Advanced/Mastery classes you want your characters to be and set their goals to get there, you can auto-instruct. You should DEFINITELY set two people to do a chore each week to help with that, it's a good way to train them in Flying, Heavy Armor, or Riding. There's literally no downside.
You can battle every weekend except for one if you want but it's the dumb way of going about the game. Do everything you can to raise your professor level as fast as you can is the smart way. That means gardening every week you explore, fishing whenever you have bait, answering letters in the chapel- those things cost no activity points and raise your professor level and/or build supports. If you ARE battling every Sunday except one, be sure that you look at the calendar and make sure that the following Monday after you explore has a green check mark for instruction. Because yes, YOU NEED TO FUCKING MOTIVATE YOUR STUDENTS. I don't know how many times I have to tell you, Bebpo. If you're auto-instructing them and they have no motivation they're not learning anything extra. If you don't like the dumb life sim parts of the game, well, this wasn't the game for you. Go sell it and quit complaining.
As far as recruiting, you can not recruit any other students but I really, really recommend you get the teachers/knights that are available at Lv 15. If you're Black Eagles, you are actually mostly ok as the game progresses as far as party composition as long as you get both Linhardt and Dorothea to rank C in Faith so you have two Physic users. Being able to heal from a distance with two units is pretty much a luxury you don't want to be without just in case. As for the rest of your team- they're adequate now but keep in mind you're 1/6 of the way through the game. By the 2/3 mark, my Ferdinand and Bernadetta were not really useful as battle units anymore, and Caspar had been useless since about where you're at. (I hate glass cannon units in FE though.) Petra also gets less useful as the game wears on because her STR growth sucks and she's just not doing damage anymore.
Felix- just better than Petra
Leonie- just better than Ferdinand
Alois- just better than Caspar
Shamir, Manuela, and maybe even Hanneman are likely better than Bernadetta depending on growth rates as a ranged option.
If you stop using your starting house units in battle they still show up in cut scenes as long as they're alive, though. If that matters to you.
