I'd assume that a TM does not accept any input if for every input $w$ it either loops or halts and rejects.
The general approach to show some language is undecidable goes as follows:
- Assume that there exists a machine $M$ that recognizes the language you are given
- Take some language that you know is undecidable
- Create a machine $N$ that using $M$ solves the undecidable language
- Since the language is undecidable, you have reached a contradiction
Do you see how being able to answer the question "is it true that machine $T$ doesn't accept any input?" would help you answering the question "does machine $T'$ accept $\epsilon$?"?
Hint: Sometimes it is useful to create a machine that ignores its input!
About semi-decidability: There's a theorem that says that a language is decidable if and only if it is semi-decidable and its complement is semi-decidable. Once you prove that your language is undecidable you can easily check that its component is semi-decidable, which immediately gives you that the language itself cannot be semi-decidable.
Regarding Rice's Theorem: Generally I'd say it is a very useful theorem, but I would recommend to solve some undecidability exercises using just pure contradictory reasoning, since it allows you to understand the concept of reducing one problem to another better, in my opinion :)