Learn Dutch: Ik vind dat een heel goed idee. - I find that a very good idea.

Because idee is a het-word (het idee). When a het-word is preceded by een (a) or no article at all, the adjective describing it does not get an extra -e at the end.

Is dat used here as a connecting word, like in 'I think that...'?

No, here dat is a demonstrative pronoun pointing to a specific thing (meaning 'that plan' or 'that suggestion'). If you wanted to use dat as a connecting word, you would say Ik vind dat het een heel goed idee is, which adds the word het (it) and pushes the verb is to the end of the clause.

Could I use denk instead of vind in this sentence?

In this context, you need vind. The verb vinden is used to express your personal opinion or judgment about something. The verb denken is used for stating facts, making guesses, or the literal act of thinking in your brain.

How would I rewrite this sentence in the future tense using gaan?

You would add the correct form of gaan next to the subject and move vinden to the very end of the sentence as an infinitive verb. The sentence becomes Ik ga dat een heel goed idee vinden. (I am going to find that a very good idea).