The whole part is no final product which goes to an external customer. It's a part which will get into the final product later.
That is OK. The final assembly is your customer.
Your customer has a demand rate, that is the takt time for the operation you are studying.
1) I'm not sure if I have the right product family. Actually I have 2 production branches which will become one later.
[...]Since I think it's better to first look at the value streams separately
Yes. treat each of those branches as a separate value stream. Think of them like two rivers that later flow into one. Map them that way as well.
I don't know how to choose the product family. Is it better to choose only the product family for the final gear wheel or should I choose a separate product family for each detailed value stream?
For Example, not every coil will be assembled to the gear wheel. So after the production of the coils, many coils will continue in different ways and not all to my focused final gear wheel.
The coils are a separate value stream, with multiple customers - just like a little factory would be. The aggregated demand of those customers forms the takt time for the coils.
2) My second and most important problem is that the company has only machines which process many parts. They use only shared resources.
So beside the product family there are always many other parts which are to be processed at the machines. So actually I don't have a real downstream process.
How can I take account of the other parts when creating the current state map especially concerning to the capacity of the machines. For example, due to the customer we need to produce about 7 final gear wheels a day. So I get a takt time of 60 minutes, but the maximum cycle time of a machine is about 30 minutes. I would know that I have free capacity. Of course! This machine has to be used for other parts as well. So how can I create a current map with these restrictions and provide a meaningful conclusion?
The machine has a total operational takt time, so you have to map that as the requirement - ALL of the demand on that machine, not just your piece of it.
Your map at that point needs to show how the information flow(s) distinguish between the parts you are concerned about from the others that are just passing through.
What are the relative batch sizes?
How often does that machine serve YOUR value stream, vs. the others?
That is, in turn, going to drive batching and queing in front of, and after, the machine.
The faster that machine can cycle across serving its various customers, the smoother the flow through that bottleneck.