If you assume that an undecidable statement in a consistent axiomatic system is true (or false), will that new system also be consistent?
For example, does $\mathsf{ZF}$ being consistent imply that both $\mathsf{ZFC}$ and $\mathsf{ZF}\lnot \mathsf{C}$ are also consistent because the axiom of choice is undecidable in $\mathsf{ZF}$?
I believe that the answer is yes, because if assuming that it was true resulted in an inconsistent system (a contradiction), then it would have to be false, therefore not undecidable.
I would use "disprovable," not "false," in your final sentence, but other than that, yes, you are correct. Saying "$\varphi$ is undecidable in $T$" is just another way of saying "Both $T\cup\{\varphi\}$ and $T\cup\{\neg\varphi\}$ are consistent," and this is due to the deduction theorem (https://en.wikipedia.org/wiki/Deduction_theorem). There are logics for which the deduction theorem fails, so this is not trivial (although there can be no complete logics for which the deduction theorem fails, for obvious reasons); see https://mathoverflow.net/questions/132268/deduction-theorem.