"Non-trivial properties" for Rice's theorem are properties of the language, not of the machine itself - for instance, it's decidable how many states a TM has, which is certainly a non-trivial property! See Perplexed by Rice's theorem for a bit of a clarification on this. Now, one of your two properties is a property of the language, while one is a property of the machine; this should allow you to apply Rice's theorem in one case but not the other. Note that this isn't conclusive evidence that the other case is solvable, but it's guidance that you might try looking for a positive answer there.
(Note that this is meant to be a very loose hint just to guide you; I can certainly go into more specific details if you'd like...)