full time webdeveloper here with a side hustle as "FREELANCE SOFTWARE DEVELOPER". 

>Seeing as I want the option to start or stop my side hustle when I want, I was thinking to register as a freelancer since I want to test the side hustle first. Does it make sense to do this as a freelancer? Or does the Gewerbe option offer other advantages if any?
In germany being a "freelancer" is not like you can sign up as being a freelancer. Your tax office determines whether you are allowed to be a freelancer or if you have to sign up as a "commerce"(gewerbe).  

It's not easy to answer that question, as it differs from state to state. 

I wasn't allowed to work as a "freelancer" per se and had to sign up a commerce (gewerbe). But you can create your gewerbe as a "small business" where you don't have to tax your invoices. "Gemäß §19 UStG ist in dem ausgewiesenen Betrag auf dieser Rechnung keine Umsatzsteuer enthalten". Sorry i wont translate that one :D

>2. Some customers need some smaller bug fixes and maintenance done on their website and I am not sure how to bill them for these tasks. Especially, since invidually they're small. Is an hourly rate or billing them a fixed price the fair way to go?

That depends only on you. We can't tell you how you should charge your customers. 
But let me tell you this, even an fixed price is an hourly rate. BUT you already did the calculation on how long it will take you to finish the "bug", "feature" or "project". Normally the time it took you to do the calculation should already be in the "fixed price" you tell them.

If you have any other questions or you might need assistance in you're first steps, HMU