Deteremine if Language is in $R$ or $RE$
The language is in RE: simply generate all inputs $x$  with $|x|\le 10000$ (they are finitely many), and simulate $T$ on all inputs in parallel. Whenever one simulation halts, accept.
The language is not in co-RE, as otherwise it would be decidable and we could solve the Halting problem.
Indeed, to decide whether a Turing machine $T$ halts on empty input, we could create a new machine $M$ that simulates $T$ if $x=\varepsilon$ and does not halt otherwise. Then $M \in L$ if an only if $T(\varepsilon)$ halts.