Here is an argument that HALTALL TM (from Question 2) is Turing recognizable: We can contruct a TM S to recognize HALTALL TM as follows:
Given an input M, S will simulate M for one step on the first string in {0,1}*, then simulate M for two steps on each of the first two strings in {0,1}* , then simulate M for thee steps on each of the first three strings in {0,1}* and so on. If at any point M halts on a string w after some number of steps, we remove w from the simulation and continue with the other strings. If M is in HALTALL TM then eventually M will halt on all strings in {0,1}* , so all the strings will be removed from the simulation. At that point S will halt in an accept state. If M loops on any w, then w at some point be added to the simulation but will never get removed, so S will loop (thereby never accepting M). Clearly S accepts M if and only if M is in HALTALL TM. So HALTALL is Turing-recognizable.
Is this argument sound?