Although this question has been solved by Arjun sir using definitions of recognizability and decidability.
But many might be wondering how to apply Rice's theorem on this problem.
$L = \{⟨M⟩∣M \,\,\,is \,\,\, a \,\,\,TM\,\,\, and\,\,\, there\,\,\, exist\,\,\, an\,\,\, input\,\,\, whose\,\,\, length\,\,\, is\,\,\, less\,\,\, than\,\,\, 100, on \,\,\,which\,\,\, M\,\,\, halts \}$
We cannot directly apply Rice's theorem here because "Halting" does not necessarily mean "Acceptance".
So, we first need to do some reduction from the given "Halting on some input of length $<100$ " problem to "Accepting some input of length $<100$ ".
This is easy because we can simply make the Reject state as Accept state in the given TM M.
In General :
Halting Problem : $ Halt_{TM} = \{ <M,w> | M \,\,\, is \,\,\, a\,\,\, TM\,\,\, and\,\,\, M \,\,\,halts\,\,\,on \,\,\, input \,\,\,w \} $
Acceptance Problem : $ A_{TM} = \{ <M,w> | M \,\,\, is \,\,\, a\,\,\, TM\,\,\, and\,\,\, M \,\,\,accepts\,\,\, input \,\,\,w \} $
Reduction from "Halting problem to Acceptance Problem" :
Reducing $Halt_{TM}$ to $A_{TM}$ : Given a Turing machine $M$ and an input $w$, let $M′$ be the machine obtained from $M$ by
marking all rejecting states as accepting. Then
$⟨M,w⟩∈Halt_{TM}\,\,\, iff \,\,\, ⟨M′,w⟩∈A_{TM} $
Reduction from "Acceptance problem to Halting Problem" :
Reducing $A_{TM}$ to $Halt_{TM}$ : Given a Turing machine $M$ and an input $w$, let $M′$ be the machine obtained from $M$ by
getting to an infinite loop at every rejecting state (by making every rejecting state a looping state). Then
$⟨M,w⟩∈A_{TM} iff ⟨M′,w⟩∈Halt_{TM}$
https://cs.stackexchange.com/questions/70989/proving-that-the-halting-problem-is-not-turing-reducible-to-the-acceptance-probl
So, we can reduce the "Halting on some input of length $<100$ " problem to "Accepting some input of length $<100$ " by simply making the Reject state as Accept state in the given TM $M.$
Now the language we consider is :
$L' = \{⟨M⟩∣M \,\,\,is \,\,\, a \,\,\,TM\,\,\, and\,\,\, there\,\,\, exist\,\,\, an\,\,\, input\,\,\, whose\,\,\, length\,\,\, is\,\,\, less\,\,\, than\,\,\, 100, \,\,\,which\,\,\, M\,\,\, accepts \}$
Now, On this problem/language $L'$ we can apply Rice's theorem.
The Property $P$ here is "Accepts some input of length $<100$" which clearly is
1. Non-trivial property of RE languages : $L(T_{yes}) = \{ ab \}, L(T_{no}) = \{ a^{100}b^{100} \}$
2. Monotonic property of RE languages : If any RE language satisfies the property P i.e. it contains some string of length $<100$ then All its superset languages will also contain that string and hence, all superset languages will satisfy the property P.
Non-trivial property of RE languages implies $L'$ is Non-Recursive/Undecidable. Hence, $L'$ is Not-REC, as we can reduce $L'$ to $L$ (by reducing acceptance problem to halting problem) so, it makes $L$ undecidable.
But monotonic property doesn't imply anything about recognizability so we have to apply the definition of recognizability here to get the final answer which Arjun sir has explained in the following post :
https://gateoverflow.in/7427/which-following-languages-recursively-enumerable-language