Learn Dutch: We kunnen straks in de tuin bijkletsen, toch? - We can catch up in the garden later, right?

In this sentence, toch acts as a question tag, just like "right?" or "can't we?" in English. You put it at the end of a statement when you assume something is true and you want the other person to confirm it.

Could I use hè or of niet instead of toch here?

Yes, you could use hè. Both toch and hè ask for confirmation, though toch sounds a bit more confident that the answer will be yes. Using of niet (or not) would also work, but it sounds a little more direct or challenging than toch.

What kind of verb is bijkletsen and why is it at the very end?

Bijkletsen is an informal verb that means "to catch up" by chatting. It is pushed to the very end of the sentence because the second position is taken by the conjugated modal verb kunnen (can), which always sends the main infinitive to the end.

Why is the order straks in de tuin rather than in de tuin straks?

This follows the classic Dutch "Time, Manner, Place" rule. Because straks (later/shortly) tells us the time, it naturally comes before in de tuin (in the garden), which tells us the place.