Complete problems and universal simulator machines
There is a universal simulator for $\mathsf{DSPACE}(o(\log \log n)) = \mathsf{DSPACE}(O(1))=\mathsf{REG}$. Namely, $U_{REG}(p,x)$ treats the first input $p$ as the description of a DFA, and then runs that DFA on input $x$. The universal simulator, however, is not itself in $\mathsf{DSPACE}(o(\log \log n))$---the $U_{REG}$ I just described uses space essentially $O(p)$---so I don't know about complete languages in this case.

More generally, complete problems and universal simulators are related, but not equivalent. For one, a universal simulator for a class $\mathcal{C}$ need not be (and in general won't be) in the class $\mathcal{C}$ itself, as in the above example. Your example of "polynomially-hampered universal TMs" for $\mathsf{P}$ exhibits this distinction even better: let $M_1, M_2, \dotsc$ be an enumeration of poly-time TMs, where $M_i$ is hampered to use time at most, say, $n^i + i$. Then let $U(i,x) = M_i(x)$. $U$ is a universal simulator for $\mathsf{P}$, but there is no single polynomial that bounds the runtime of $U$, so $U$ itself is not in $\mathsf{P}$ and hence not complete. 

Furthermore, there cannot be a single polynomial that bounds the runtime of a universal simulator for $\mathsf{P}$: if the universal simulator ran in time $n^k$, it would never be able to decide a language which required more time, say, $n^{k+1}$. We know that such languages exist by the Time Hierarchy Theorem.

However, the other direction usually holds. That is, if a class has a complete language under essentially any of the natural types of reductions, then it has a universal simulator. As an example, consider $\mathsf{NP}$ and its standard complete language SAT. Then we construct a universal machine for $\mathsf{NP}$ as follows (we knew how to do this for $\mathsf{NP}$ already using polynomially-hampered nondeterministic machines, but you'll see how this construction generalizes to any class with a complete language). $U(i,x) = SAT(M_i(x))$ where $M_i$ is the $i$-th poly-time deterministic TM, as above. Here we are using the $M_i$ to list out the possible reductions to the complete language, rather than the deciders of languages themselves.