Categorising problems into complexity classes
Your reasoning seems correct, but you are missing a crucial step once the second bullet point is done with: You check that your problem is also complete for the class $C$ you have found for it. For instance, if you have a problem $P$ and show it is reducible to a complete problem in $\mathsf{PP}$, then you are not done yet. For all we know, $P$ could be $\mathsf{PP}$-complete, but it could also be in $\mathsf{NP}$, or $\mathsf{BPP}$, etc. Therefore, to finish classifying $P$, you should be able to find a class it is complete for—if possible; after all, there are classes which are not closed under (poly-time many-one) reductions—, or provide some good evidence why it could not be in a class which is known (or conjectured) to be strictly below $C$.


  If I cannot find a reduction and I do not have an algorithm, what is the "default" complexity class?


Assuming your problem is decidable, the "default" complexity class is $\mathsf{R}$, the class of recursive languages. Improving on this is only possible by producing an algorithm whose time complexity you can prove or, alternatively, proving a reduction to some other problem for which this is the case; that is, essentially what you wrote in your question. There is no such thing as a "class of problems which cannot be solved until we find an algorithm" in complexity theory; that would be the domain of computability theory (i.e., problems which are not known to be decidable).