Translation of ordinary language assertions into logical language is sometimes difficult. Certain subtleties and connotations simply do not translate. There are always a number of logically equivalent translations. And in a fair number of cases, there can be two logically inequivalent translations that are each sensible.
Question 1: This is right. There are other right answers, such as $V(c) \land \forall x K(c,x)$.
Question 2: As usual there is more than one translation. We want to say something closer to $\exists x (P(x) \land \forall y((P(y)\land \lnot(y=x))\implies \lnot T(y,x)))$.
The implication symbol can be avoided in the usual ways, since $A \implies B$ is logically equivalent to $\lnot A \lor B$ or $\lnot(A \land \lnot B)$.
Your version does not contain the negation, and is in many other ways not close. It seems to say among other things that everybody ("$y$") is a politician. Also, the sentence needs to say that this bad politician $x$ is not trusted by any other politician. So we need to make sure, by using the $\lnot(y=x)$, or in some other way, that we do not claim that this bad politician $x$ does not trust herself. This question is harder than the other two.
Question 3: The sentence $\forall x \exists y (P(y)\land \lnot T(x,y))$ should work. Read this as "for any person $x$, there is a politician $y$ whom $x$ does not trust."
Your version is very far from saying the desired thing. Somehow you have the $\lnot$ inside the $T$ relation.
There is a case to be made for $\forall y(P(y) \implies \exists x\lnot T(x,y))$.