Question 1:
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Many people failed to define a language.
A comon mistake is to confuse TMs and languages, and to try to prove that a TM decidable or undecidable.
Many wanted to prove L decidable by proving L and L^c both recognizable. (The idea is ok. But the proof is much harder.)

Question 2:
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Only a function can be computable or not computable. Only a language can be decidable or undecidable.
A TM  can be a decider/recognizer to decide/recognize a language.
Many students fail to identify the difference among the three concepts.
Most students fail to supply enough proof for the decider of E_TM, such as f is monotone increasing.
Many students try to prove by a reduction.
Their logic looks like "if an undecidable L1 is reducible to L2 with function f and L2 is undecidable,
then f is computable". This logic is not correct.
The idea of reducible is "if an undecidable L1 is reducible to L2 with function f and f is computable,
then L2 is undecidable".
The two parts, "f is computable" and "L2 is undecidable", cannot be swapped.

Question 3:
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Most students notice that the new TM can go to left of leftmost cell of input and try to avoid it.
However, the solutions are not perfect. Generally, 2-3 marks are reduced at this point.
No matter whether a contradiction or a reduction is used in this question,
the proof always has to be double way.
Some students supply a very insufficient proof.

Question 4:
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I am surprised that so many students misunderstand the problem.
Their function looks like f(<M,w>) = <M', 1w>.
In this case, 4-5 marks are reduced based on their proof.