Consider the following circuit: Now imagine that I add a 3A current source: Is the following equation correct? $$I_1=I_0+3$$
No, I1 is by definition 3 amps. You cannot put ideal current sources in series just as you cannot put ideal voltage sources in parallel. Doing either one causes a contradiction.
What would happen in real life if I did add the current source? Will the circuit explode? Or will it just become non-ideal? Will it convert from current source to voltage source? How do voltage sources and current sources even work? If we use the analogy of comparing electric potentials to gravitational potentials, are voltage sources like ladders or slides?
Real current sources have a range of voltage that can be applied across them and still work. This voltage range is often called the "compliance" range. If you try to operate a real current source outside of this range, it no longer is a constant current source, and usually will put out less than the intended current. If you stack 2 current sources that have different current ratings, generally the higher current one will go out of compliance, and will output only the current that the lower current source is rated at. Thus, both current sources in series are passing the same (lower) current. You can see this for youself -- look up common circuits for a current sink and a current source. Then stack the two schematics and figure out what will happen... (fun exercise) Sorry, I have no idea what that means.
In real life, current sources are NEVER ideal so the question you seem to be asking (what would happen if I put ideal current source in series) doesn't make sense. Berkeman has explained what will likely happen if you put real current sources in series.
I see the circuit posted as a current source in series with a voltage source. There is no reason that cannot be done.