Short answer: no, you cannot take your bicycle into all German trains (but you can probably reach any destination in Germany).
Many trains in Germany carry bicycles. This is particularly true for local trains, but also most intercity (IC) trains. However, the most common train for long-distance travel is the high-speed InterCity Express or ICE. Usually, on ICE, you can not take a bicycle¹.
On the website of the German railways, you can search specifically for trains carrying bicycles. Select carriage of bicycles required in the advanced search dialogue. Taking a bicycle costs €6.
For example, from Norddeich to Basel, if you need to take a bicycle, the fastest journey is 12 hours, 13 minutes, with 5 changes. If you don't need to take a bicycle, the fastest journey is 8 hours, 51 minutes with 1 change.
So in a nutshell: you can probably reach any station in Germany while travelling with a bicycle, but bicycle cannot usually be taken on high speed trains. Therefore, travelling a long distance with a bicycle will take considerably longer and likely require more changes than travelling without one.
(As for your second question; I don't know if there is such a service)
¹An exception is when the bicycle is folded or otherwise taken apart; a folded bicycle can be taken as ordinary luggage on any train. I think there are some ICEs that take bicycles, but those are currently the exception and not the rule. This might change in the future; reportedly newer ICE trains do have some space for bicycles. There are reports of people successfully taking bicycles on the (regular/older) ICE, but formally this is against the rules. Your experience may vary depending on how full the train is, on train staff, and on how convincing you are.