I dont have an answer to this, but this is too long for a comment.

As for your fist question, one thing to check is whether the resulting locally ringed space is a scheme. This doesnt happen always, as Balmer showed in his paper Spectra, spectra, spectra Proposition 9.7.

He concretely shows that the locally ringed space associated to $SH^{fin}_{(p)}$, the topological stable homotopy category of finite spectra localized at p with smash product as the monoidal structure, is not a scheme. He does so by checking that if it were a scheme it would be the spec of the ring of global sections, which he shows is local and then compares the number of points of the underlying space.

I believe that you could abstract a bit how this counterexample goes and arrive to a statement with a somewhat contrived test to check whether the space is a scheme. It is my impression that the general suspicion is that triangulated categories of topological nature would hardly give you a scheme, but then again I dont know of very general theory on this idea.

However even if the space is a scheme I think it wont guarantee you the TT-category is a derived category. A semisimple abelian category is triangulated, so the usual tensor product on something like $k-Mod$ will yield $Spec(k)$ under Balmer's reconstruction, but $k-Mod$ is not ( I dont think?  ) equivalent to a derived category of a scheme. But this example looks a bit extreme.

As for your second question this seems even tougher, the extreme case I can think of is when the derived category already determines the scheme as the Bondal-Orlov reconstruction setting, some discussion on the tensor products that arise in this situation was discussed in this MO question although this doesnt address your question per se.

I don't have an answer to this, but this is too long for a comment.

As for your fist question, one thing to check is whether the resulting locally ringed space is a scheme. This doesn't happen always, as Balmer showed in his paper Spectra, spectra, spectra Proposition 9.7.

He concretely shows that the locally ringed space associated to $SH^{fin}_{(p)}$, the topological stable homotopy category of finite spectra localized at p with smash product as the monoidal structure, is not a scheme. He does so by checking that if it were a scheme it would be the spec of the ring of global sections, which he shows is local and then compares the number of points of the underlying space.

I believe that you could abstract a bit how this counterexample goes and arrive to a statement with a somewhat contrived test to check whether the space is a scheme. It is my impression that the general suspicion is that triangulated categories of topological nature would hardly give you a scheme, but then again I don't know of very general theory on this idea.

However even if the space is a scheme I think it wont guarantee you the TT-category is a derived category. A semisimple abelian category is triangulated, so the usual tensor product on something like $k-Mod$ will yield $Spec(k)$ under Balmer's reconstruction, but $k-Mod$ is not (I don't think?) equivalent to a derived category of a scheme. But this example looks a bit extreme.

As for your second question this seems even tougher, the extreme case I can think of is when the derived category already determines the scheme as the Bondal-Orlov reconstruction setting, some discussion on the tensor products that arise in this situation was discussed in this MO question although this doesn't address your question per se.