Learn Dutch: Dat lijkt me leuk. - That seems nice to me.

In Dutch, the verb lijken (to seem) doesn't need a preposition like "to" when you're saying who it seems that way to. You can just put the object pronoun (me) directly after the verb.

Why use me instead of mij here?

me is the unstressed version of mij. In everyday speech, you use me unless you really want to emphasize that it seems nice to you specifically, as opposed to someone else. Here, it's just a general statement, so the unstressed me is perfect.

Could I say Het lijkt me leuk instead of Dat lijkt me leuk?

Yes! Both are very common. Using Dat (that) just points back a little more strongly to whatever specific idea or plan was just mentioned. Het (it) is slightly more neutral, but they are often used interchangeably in this kind of reaction.

Can I use lekker or mooi instead of leuk here?

You could, but it changes the meaning! leuk means fun or enjoyable, like an activity or idea. lekker is usually for physical sensations or food, and mooi means beautiful. So for an idea or plan, leuk is the most natural choice.