I am US citizen that moved to Germany with my wife who is a German citizen. She works in Germany.I am new to the country and would like to apply for for a remote US job. Would I need any special documents? Would I need to pay any extra taxes? What else should I know about this situation? Help please!
Your residency permit includes a work permit. You can take on any type of work you want, in this regard you are no different from a German citizen. That includes working for foreign companies, which is a lot easier for you because you also are allowed to work in the US due to your citizenship.
If you happen to find a US company that has a branch located in Germany, you can work for the branch, that is a regular German employer. However, I assume you mean you actually want to work for a US company based in the US, remotely. You can do that, too.
You have, however, to pay taxes in Germany and in the USA. The most likely way this goes over is you pay your US taxes normally, then declare your income for the German taxes and deduct the taxes you paid in the US. This is possible because the US and Germany have a double taxation treaty (well, one that prevents double taxation). As taxes in the USA are lower, you will probably have to pay regular taxes in the US and some additional taxes in Germany, too. So in the end, you will have to pay the higher (German) amount of taxes, part of it to the US and part of it to Germany.
I strongly advise you to get a tax accountant for that, especially to do all the number crunching on whether it would be better for your wife and you to file jointly or separately and advise your wife which tax bracket she should apply for, for her own tax declaration.
The following is my experience, you need to research this on your own, because a lot may have changed over the years and I'm not even sure if my experience was due to the legal framework or my specific providers policies:
If your wife has a regular public health insurance, right now, you are on it. If you don't work, all your health expenses are covered by your wife's insurance (assuming you tell the provider). Make sure you are on it first for a few months, before you get a job on your own. That will probably happen automatically when you move and need to adjust and find a new job or are officially unemployed while your old employer figures out the paperwork they need to do to keep you employed overseas.
When you get your own job, whether that is in Germany or elsewhere, you need your own health insurance, you can no longer claim to be your wife's dependent with no own income. Health insurance is mandatory for residing in Germany, it does not matter where you are employed.
Health insurance fees are calculated off of your income up to a certain limit. German employers are required by law to pay half of their employees health insurance fees. Since you don't have a German employer, you will have to pay both halves yourself, as if you were self-employed.
As a remote worker in the US, you will have a salary high enough that you pay the maximum amount. The maximum amount per month is regulated somewhere but it should still be ~800 EUR per month. That sounds a lot, but remember Germans with a German employer only pay half of that. Now when you change from being your wife's dependent to a independent customer of the health insurance provider, you have to fill out a four-plus-page form about pre existing conditions. The top of the form has a little checkbox that says "I will pay the maximum amount per month". Since you have to tick that anyway, that checkbox skips the whole form. Whatever ails you, you got coverage if you pay them the maximum amount.
And here is why I said make sure you are on your wife's insurance as unemployed dependent first: an insurance does not have to take you. But since you are already their customer, they cannot deny transferring you from one state into another (dependent vs independent paying member and maybe back if you ever lose your job). Again, that might be a regulation, or maybe just a policy of my provider or maybe it doesn't matter because it changed in the last 5 years. You have to find that out. But make sure you find it out.