Yes, you can do the Nebengewerbe/Einzelunternehmen while having a full time job. You should create an additional bank account and use that one for your side business, which will make it easier for book keeping (otherwise you have to open your private bank account to the tax authorities). You can also get a VAT ID, if you do EU business outside of Germany.
Freelancer is something else as you might think, some jobs fall in the category of "Freiberufler", which is some creative jobs, etc... All others are simply "Gewerbetreibende".
You can also found a UG/GmbH on the side, you don't need to do it fulltime. You should look in your contract though if there is anything that prevents you from doing so.
Sadly I can't suggest anyone since I had to take care of everything myself as well, I simply work together with a regular tax guy. I am based in Germany as well and started basically the same way as you.