Answer : 1
$S_1$ : "There exists no algorithm for deciding if any two Turing machine M1 and M2 accept the same language."
It is the standard "Equivalence problem" for the Recursive Enumerable languages and It is Undecidable. Hence, No Algorithm possible.
$S_2$ : "The problem of determining wether a Turing machine halts on any input is undecidable."
Yes. It is the standard "Halting problem of TM". It is Undecidable and Hence, No Algorithm possible.