The weight of the ply won't be a factor if adequate drawer slides are used. The 1/2" ply will work fine though, assuming you have sturdy joints. I'm not sure what part you are asking about reinforcing, but it's not necessary-again, assuming the drawer box joints are strong.
Another place that can be overdone is the drawer bottom. I've seen drawers of a similar dimension to the one you're planning that used 1/8" ply for the bottom, and they are holding up just fine, even loaded with heavy pots and pans. When the drawer bottom is captive on all four sides, it's incredibly strong. It won't sag like it would if it were a shelf, supported only on the ends. Nevertheless, I never have, nor ever would, built a drawer with that thickness on the bottom. What I'm leading up to is that it's often assumed that you need a thick piece to support the contents of the drawer when it will be holding heavy objects. There's no harm in using a 1/2" ply for the bottom, but 1/4" will be adequate for any load that 100lb drawer slides will support.
That's part of the balance of design. Don't build a drawer box to support more than your slides are rated for. Again, it's no big deal, but why do it?
I use undermount slides exclusively for my cabinet construction. It's a matter of aesthetics though, not strength. Undermount slides will do nothing to add support for the bottom of the drawer since it's already being supported by being captive in the groove in the drawer sides.
Getting back to the joints, those are the most likely points of failure. If you try to construct them with pocket screws (which I see so many Instagram and Youtube hacks do), it's going to fall apart eventually. Stick with half-blind dovetails or, at a minimum a drawer lock joint. That's what will give it lasting strength.
I never glue in my drawer bottoms. It serves no purpose. The back will never bow enough to let the drawer bottom pop out. Think about it. To have that happen, the rear joints would have to fail, because bowing the back out will also pull in the sides, which is prevented by the drawer bottom. Sure, it's plywood, and movement is not an issue, but why bother?
Think about the physics of the load. If the contents are not fluid, then the load will all be vertical. I don't care how many pots and pans, or cans of vegetables you load in there, there will never be significant pressure on the walls of the drawer to cause failure. Yes, if you fill it with 100 lbs of lead shot, then the walls will all be strained, but again, if it's properly constructed, it will withstand even that load.
To sum up, strong joints are the essential element to construct a drawer box that will not fail over time.