Getting solid firing pin impacts on the primer is not a problem with striker fired weapons, as the increase in drag on the system in water as opposed to air is minimal. Some hammer fired weapons can be problematic as you have the drag of the water on the hammer itself before it even impacts the firing pin. That is then complicated by the harder cups on mil-spec primers that are intended to reduce accidental discharges due to potential slam fires from inertial firing pins under high cyclic rates of fire.
The powder used in cartridges contains its own oxidizer, so it's a closed system that will function just fine underwater or in a vacuum. Sealing the primer to case head interface and the bullet to case neck interface is not a bad idea, but it's also not necessary over the short term as even normal non-specially prepared ammunition is snug fitting enough to be waterproof over the time span required. Sealing cartridges is done mostly to improve reliability over the long term in anticipation of cartridges being stored under far less than perfect storage conditions. However, on non-roll crimped large caliber pistol rounds where neck tension may be minimal, a sealer can help bond the bullet to the case and/or create a lip of sorts to prevent set back and resulting leakage due to water pressure at depth.
As noted above, you need to fire the weapon with the barrel fully flooded, or fully drained. Firing the bullet in an air space in the barrel with the bullet then impacting a slug of water in the barrel is bad news. At best it will ring the barrel, and at worst the barrel will fail, so it's important to fully flood the barrel when entering the water. Similarly, it is important to fully drain the barrel when exiting the water. Long barrels in small calibers with a tight seal of the cartridge and/or bolt in/to the chamber can be problematic as they can retain water in the barrel in much the same way you can retain water in a drinking straw by placing your finger over the hole at the top.
A few weapons designed for use in and around water are designed so that water will drain from the barrel on exit, and those that are not need to have the bolt manually retracted enough to allow the barrel to drain. In that regard a weapon that fires from an open bolt seems ideal, but the downside is that the whole bolt has to move forward to fire the weapon and that reduces the bolt speed and the impact of the fixed or inertial firing pin on the primer, so it's a mixed bag. In any automatic or semi automatic, you may need to adjust the recoil spring forces to ensure adequate bolt speed to fully cycle the weapon, and you need to ensure that there is not some form of hydraulic buffering occurring behind the bolt.
As an aside, if you want to have fun with hydraulics and have a friend who shoots an AR-15 - put a couple ounces of grease in the bottom of the buffer tube assembly. You want enough not to be noticed when the action manually cycled, but enough that when he fires it, he'll be on the receiving end of a .223 powered grease gun (provided it has a buffer tube retaining screw with the drain hole in it).
