'that make me' or 'which make me' and when to omit 'that'
Solution 1:
"My teaching... are not the only factors which/that make me a suitable applicant." => 'which' and 'that' are relative pronouns (they stand for 'factors'). Since they are the subjects of the verb 'make', you cannot omit them. Both are correct. 'Which' is more formal.
In sentences 2, 3, 4, 'that' – whether it's a relative pronoun or a relative conjunction – is not the subject of the verb that follows. You can omit 'that', if you like... Omitting 'that' will make your sentence slightly more informal.