I just approved an edit, but upon revisiting the post, I saw some other things I wish I had improved upon. Since "approve and improve" is a review option, can I make improvements without having to wait for a second reviewer? I haven't found a way to do that.
You do need to wait for the second reviewer to clear the edit. You could bring a post with a pending edit to the attention of the community here on meta to get it out of the review queue more quickly.
You can always improve a post, you just need to wait for it to complete the review process so that it doesn’t have pending edits.
You may want to bookmark the post or add it to your favorites to make it easier to find. You may find that some has made edits similar to what you were planning by the time you get back to that post but it doesn’t really matter who makes the edit as long as the post is improved.
You can search your favorites by adding “infavorites:mine” to your search terms.
As far as I know, it's impossible to do this. If you voted to Reject but want to change to Reject and Edit, it's doable, although not ideal: just go to the post's revision history and click "edit" on the top revision, which will automatically override the suggestion once you finish it.
This does have two problems. First, without installing a userscript for the purpose, it's rather awkward to get to the edit screen reliably. Secondly, unlike Reject and Edit, it doesn't tie into suggested edit rejection stats and automatic bans.
But in any case, you can't do any of this with a suggestion you approved, since it would just reject it. You'll just have to leave the post open and wait for the automatic change detection to let you know when you can edit it.