1. A body of water, smaller than a river, contained within relatively narrow banks?
2. The thing you push around the grocery store?
A trolley (not a shopping trolley, which is something different)
3. A metal container to carry a meal in?
Ours were never metal, but the plastic equivalent is a lunchbox.
4. The thing that you cook bacon and eggs in?
A frying pan.
5. The piece of furniture that seats three people?
Couch or sofa. It used to always be 'couch', but our current one is a sofa-bed; I've never heard it called a couch-bed.
6. The device on the outside of the house that carries rain off the roof?
A gutter, or guttering.
7. The covered area outside a house where people sit in the evening?
No UK equivalent. Such an area without a covering would be a patio.
8. Carbonated, sweetened, non-alcoholic beverages?
9. A flat, round breakfast food served with syrup?
10. A long sandwich designed to be a whole meal in itself?
Probably a baguette.
11. The piece of clothing worn by men at the beach?
Swimming trunks. (Well, on a UK beach it'd be more likely to be trousers, shirt, jumper, jacket... ;))
12. Shoes worn for sports?
Trainers or plimsolls. Plimsolls are what I wore at school for sports. Trainers I see as something for older kids or adults, not little kids.
13. Putting a room in order?
14. A flying insect that glows in the dark?
15. The little insect that curls up into a ball?
I think you mean a woodlouse.
16. The children's playground equipment where one kid sits on one side and goes up while the other sits on the other side and goes down?
17. How do you eat your pizza?
At home: with hands. At a restaurant: with cutlery.
18. What's it called when private citizens put up signs and sell their used stuff?
If they're doing it on their own property, a garage sale; if they're going somewhere else to do it with other people, a car boot sale.
19. What's the evening meal?
20. The thing under a house where the furnace and perhaps a rec room are?
Nonexistent in the UK! However in theory I would call it a basement.