I'll given you an answer to (a) and (c).
**Every likes somebody.**
$$\forall x, \exists y(L(x, y))\tag{(a)}$$
$$\lnot \forall x \exists y\Big(L(x,y)\Big)\equiv \exists x \forall y\Big(\lnot L(x, y)\Big)$$
"There is someone who everyone dislikes."
* * *
**Anyone who has heard of everybody, will be liked by everybody.**
$$\forall x \forall y(H(x, y) \rightarrow L(y, x))\tag{c}$$
Now, we will negate (c). $$\lnot\Big(\forall x \forall y(H(x, y)\rightarrow L(y, x))\Big) \equiv \exists x \exists y\Big(\lnot (H(x, y) \rightarrow L(y, x))\Big)$$
$$\equiv \exists x\exists y\Big(\lnot (\lnot H(x, y)\lor L(y, x))\Big)$$
$$\equiv \exists x \exists y((H(x, y) \land \lnot L(y,x)$$
Which can be translated: There is a someone who has heard of a person who does not like him/her.