There are a lot of ways to do this. Ask two people and you will get three answers! Here are my thoughts - and again - not the only way to do this.
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Grain orientation
All of your “rectangular” pieces should be oriented parallel to the wood grain; partially for structural stability and partially because raster engraving looks best when the laser head side-to-side travel is with the wood grain.
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Wood selection.
Personally, I think 1/16" thick is going to be a problem unless you buy hardwood, which is going to be pricey.
- 1/16" basswood. Buy at hobby stores like Hobby Lobby. At this thickness, basswood is going to be fragile. However … 3/32" basswood might work.
- 1/16" plywood. Buy at the more serious hobby stores. You might find it at the big box stores, but you might have to go to HobbyTown USA. Medium pricey. 3/32" or 1/8" might actually be cheaper.
- 1/16" hardwood (like maple). You’d have to buy that at Woodcraft or equivalent and it’s going to be the most expensive of all of them.
EDIT: If these pieces are all going to be mounted to a board and the kids aren’t going to be handling them, then 1/16" thick will probably work.
- Cut order.
- Obviously you need to raster first.
- IIWM, I’d separate the cuts into two layers so the little circles (punch holes) get cut out first. And this cut order will make more sense when you read my next comment about artwork.
- Support your wooden board on at least two edges so that it is slightly off the laser bed (and focus the laser on wherever the top surface of your board ends up). This will reduce char on the underside of the pieces. The pieces will fall out onto the laser bed when their cut is complete so cut order is important here.
- Artwork. I can’t tell from your artwork exactly how close together your parts are (no idea of scale). IIWM, I’d consolidate the “rectangular” pieces together so they share common edges and separate them with the final cut. It buys space and you don’t risk cuts that are too close charring a subsequent cut.
Here’s the concept. I’m showing only the vector cuts. After the rastering I’d cut the green lines first and then the blue lines.
EDIT: There’s another option that’s probably easier. Use your popsicle sticks and pre-cut circles and make something to hold them in position on the laser. Take a piece of cardboard (like matte board). Tape it into place on the laser so it doesn’t move. Cut sockets out of the cardboard and without moving the cardboard place your goodies in there. Adjust the laser height for the popsicle sticks; laser cut the punch hole circles and raster. Assuming that the pre-cut circles are a different thickness, you will probably have to adjust height and raster the circles separately (although you can cut all the cardboard sockets at the same time).
