Oh, there's plenty of ways to make custom planets, names, stars, systems, and have them all be your own. I've made custom planet types, art, and gradients for my home system. The only thing that doesn't work right now is custom heightmaps, but those USED to work, so keep your fingers crossed for their return. Thing is, all of this requires XML modding. Star systems, thankfully, aren't too difficult.
And how do I create ship templates for a custom race? I've been looking for it in custom race menu but it's unavailable. It seems like I have to choose a hull style and then start a game to create the ship templates for them. If I save the templates will it be always available for selection whenever I play a game as custom race as long as I have needed techs unlocked?
That is correct. Even if you want to make a custom hull-style, you can only make new ships after starting a game. If you save the templates, they will always be available for that custom race, as well as any other race that uses that hull style. Making a custom hull style is doable, but that's a long-term project because of all the work involved.
There are guides that go into detail about everything you've asked about in this forum. Some are even made by devs.