When drilling screw holes for acrylic should I make it slightly larger than the actual diameter of the screw hole so that it does not crack?
Very much yes. You're not drilling pilot holes here, you're drilling clearance holes which are completely different. See more in this previous Answer.
Depth of the screw hole for the birch plywood. Would 3/4" suffice or should I drill same length as the length of the screw since it is hard wood?
This applies elsewhere in your proposed build as well, it would actually be much better if the plan didn't call for screws to be driven into the sides of the plywood.
The hold of screws in ply is much better if driven in through the face and not into the edges. The difference is so great that it can be worth the trouble to cut smaller pieces of plywood and then glue them together in a stack to achieve a sufficient thickness. Alternatively you can use the trick posted at the end of the Answer linked to above where you drill a hole and glue in a dowel for the screw to bite into but this may require buying a larger bit (sized to the dowel used) as well as possibly a suitable saw for cutting the dowels flush with the surface. Some sanding will probably also be required to make them completely flush with the face of the plywood.
I am trying to minimise screws showing.
On the sides you can do what's called a counterbore (see this previous Answer) and plug those with wood or fill with wood filler to hide the screw heads.
Will this setting be strong enough? I am concerned about 2mm acrylic panel wobbling especially.
Four screws holding the acrylic in place is very likely not sufficient to prevent it sagging. I think the ideal for this is for a slightly larger piece to be used where all four edges would be set into a groove or rabbet/rebate, similar to how the bottom of many boxes are done. If you do want to secure it with screws a few more should be spaced around the perimeter.
Note: for the top screws holding the polycarbonate not to sit proud of the surface the drilled holes need to be countersunk and if you don't have a combination drill/pilot/countersink bit you will need to buy a countersink separately.
Countersinks are not all very well made and many cheaper ones are not very sharp so you might want to practice doing the countersinks on a scrap of the acrylic to ensure a clean and even result (e.g. oval holes are easy to create by accident). In addition to the possibly unevenness a blunter countersink will begin to heat the plastic and it could actually melt at worst. So controlling the speed of the drill is vital. Also be very careful how hard you press during both the drilling and countersinking.