Learn Dutch: Ik heb spijt van die fout.

In Dutch, "to regret" is expressed with the fixed phrase spijt hebben (literally "to have regret"). Because spijt is a noun meaning "regret", you must use the verb hebben (to have) with it.

Why is the preposition van used in this sentence?

The expression spijt hebben is almost always followed by the preposition van when you want to state what you regret. You can think of it as saying "I have regret of that mistake."

How do I change this sentence into the past perfect tense?

To say "I had regretted that mistake", you need had plus the past participle of the main verb. The main verb here is hebben, and its past participle is gehad. So the past perfect version is: Ik had spijt gehad van die fout.

Why do we use die instead of dat for "that mistake"?

The word for mistake, fout, is a de-word (de fout). Whenever you want to point out a specific de-word and say "that", you must use die. If it were a het-word, you would use dat.