Prior to Covid-19, all of my European trips over the past few years have been around 3 months. Here's what I do:
Set up all of your bills to be paid automatically. You can monitor all of this on the internet.
Get a Private Mail Box with Virtual Mail (in the USA) and have your mail forwarded to it. With Virtual Mail, the people running the PMB take a photo of the front of each piece of mail you get (magazines, advertising flyers, and the like are usually not included.) You then log into their website and look at the mail. You can then tell them to forward it, keep it, shred it or open it up, scan it and email it to you. I use it and it' great.
As for banking, set up online banking and use your credit cards and debit cards just like you would at home. No need to fiddle with trying to get a German account. With online banking you can monitor your account and move money around if you have more than one account. (Make sure you alert your banks and credit card companies of your travels so they don't think using overseas is fraudulent.) Over the past few trips, I doubt I spent more than $50 in actual currency. Most went on my credit cards.
As for health insurance, check with your own provider to see if you are covered. Although I am technically covered, I always had an annual policy for health, theft, travel delays, and medical evacuation.
Some over the counter meds like Pepto are not easily available in Europe. If you want to have these, bring them. The basic toiletries can be found almost anywhere.
As for packing, I travel with just a carry on size bag and a small shoulder bag for my electronics. I just do laundry more often. If you plan to stay in one place, look for a rental with a washing machine. Whether it's three days or three months I bring the same. The only thing I bring more of is prescription medications.
Remember, there is no porter service on European trains so you are responsible for your luggage. If you insist on taking big luggage, you could look into a luggage forwarding service like sendmybag.com. I've used them and they are very efficient.
For my phone, I have T-Mobile which offers free data in Europe although it can be slow. For calls back to the USA, I used my hotel's wifi and Skype.
As for learning German, unless you plan to visit small towns where non-German tourists rarely go, you should be fine with English. It's always nice to learn some of the local language but for a short visit the basics will be fine.
If you don't have to be in one place, why limit yourself to that one place over three months? You could experience more places.
