1. The Kids Club at Disney will feed the kids a kids meal as long as the child is at the Kids Club between the hours of 6 - 8 pm. There are usually about 4 choices (think chicken nuggets and mac n cheese type of food). The cost is the same per hour regardless of whether the child eats dinner or not. Personally, we always let the kids club feed our kids. It frees up our time and does not cost additional.
2. If you are using a private babysitter, he/she can take the child to eat as long as you give permission (e.g., whether they can leave the room, the hotel, etc.).
3. In terms of you feeding the kid before your date night, yes, you can do that too. I would look for a location near the Kids Club. However, I would recommend going to a counter-service restaurant. I would not recommend making an ADR and taking the child to eat dinner at a table-service restaurant. In general, the table-service restaurants take a longer time, and it may be strange to sit there and watching your child eating, without you yourself eating, KWIM? Also, many Disney restaurants are fixed price meals (buffets, family style, etc.) and you would be charged regardless of whether you eat or not, so if you go this route, you may want to book a regular table restaurant that is not a fixed price.