Learn Polish: Teraz lepiej to rozumiem. - Now I understand it better.

Yes! lepiej means "better" and is the irregular comparative form of the adverb dobrze (well). Just like in English, the most common adverbs often change their spelling completely when making comparisons.

Could I use lepszy instead of lepiej to say "better" here?

No, because you are describing an action (the verb rozumiem, meaning "I understand"). To describe a verb, you need an adverb like lepiej. lepszy is an adjective, which you would use to describe a noun (for example, lepszy samochód - a better car).

Why is the pronoun to in the middle of the sentence instead of at the end?

While Polish word order is flexible, short pronouns like to generally avoid being the very last word in a sentence. Putting it right before the verb sounds much more natural to a Polish speaker than putting it at the end.

What case is to in this sentence?

It is in the accusative case, because it is the direct object of the verb rozumiem (you are understanding it). For the neuter pronoun to, the accusative form looks exactly the same as its basic dictionary form.