Decidablity of Languages of Grammars and Automata
Convert G to Chomsky Normal form. This way, the only empty derivation would be the start symbol that does not appears anywhere else and thus if there is some production that may able to eventually generate itself, then the grammar is infinite. If no such production exists, each symbol will only be able to generate a finite set of strings, and then the grammar is finite. So, build a directed graph where each production is a node and each symbol inside a production is an edge targeting that symbol. If the graph has some cycle, the CFG is infinite, otherwise it is not. Hence a Turing machine for $FINITE_{CFG}$ could be construct doing exactly that, and then $FINITE_{CFG}$ is decidable.
Suppose that $FINITE_{TM}$ is decidable. Lets say that $H$ is a Turing machine that has some string as input and uses itself as an input to $FINITE_{TM}$. If $FINITE_{TM}$ returns true (i.e, $H$ accepts only a finite language), then $H$ accepts the input, which leads to a contradiction since the input set is infinite (the input's length is unbounded, so accept any possible string as input means accepting an infinite set of strings). If $FINITE_{TM}$ returns false (i.e $H$'s language is infinite), then $H$ rejects the input, meaning that $H$'s language is finite because it does not accepts any input (i.e. its language is empty), which leads to a contradiction too. This way, the supposition that $H$ exists leads to contradiction, and this supposition is based in the supposition that $FINITE_{TM}$ is decidable. So, by contradiction, we have that $FINITE_{TM}$ is not decidable.



  The same source also proves that a Context Free Grammar can be converted to a regular expression, and vice versa.


I really doubt that Sipser would state that, you probably misread or misunderstood. This would mean that context-free grammars generate exactly the same langauges as right-linear grammars. This is false; the right-linear grammars generate are a proper subset of the languages context-free grammars dp. That said, the way you tried to use regular languages to answer the questions just leads you to nowhere.

As you can see above in my proofs, the two questions are indeed two very distinct, unrelated questions. It just happens that they were worded similarly.