It's pure handwavium... but it's not completely absurd.
As noted in Christopher's answer, powerguns work via a hand-wavium reaction that generates plasma from a specially formed matrix of "stuff" (that probably qualifies as a metamaterial).
Now... we've known how to achieve partial matter to energy conversion since at least World War II. We know there are instances where a very particular application of pressure to a specially prepared physical system will result in the release of massive amounts of energy. Now, Drake's process is completely invented, of course, but it's not completely inconceivable that such a process might actually exist; he's just proposing a system not entirely unlike one that does exist, only a lot smaller.
If we take that system as an axiom, the rest seems at least somewhat plausible. Obviously, a weapon using a repeating version of this is going to need to handle a lot of energy, hence the iridium barrels (which IIRC are stated to wear out and need replacing frequently) and liquid nitrogen cooling system.
The other huge caveat is how the plasma beam maintains cohesion. Probably you need to generate a vortex with the plasma, or a long range magnetic field, or both. Again, this can be hand-waved by the metamaterial nature of the charges. This would suggest that duds might have the possibility to "bloom" excessively. (Because the barrel provides some focusing ability, it probably won't backfire so badly as to kill the firer. Usually.)
Now let's look at some of your specific questions:
Can the bullets be this easily ionized to form plasma?
First, as noted, the charges are disks, not bullets. There are no bullets involved. Per above, the plausibility of this is pure hand-wavium, but as fictional physics go, it could be a lot worse.
A powergun charge is essentially a miniature nuclear bomb, probably of significant complexity. Accordingly, it's likely much harder to manufactures the charges for powerguns than the guns themselves.
The function of the charge is to be a very precise arrangement of materials such that a relatively simple action sets off a chain of events that results in at least part of the charge flashing into plasma. If it's been made right, setting it off is relatively easy, but making the charge in the first place is probably on par with manufacturing microprocessors, or worse.
How are the plasma bullets propelled?
There are no bullets. When the powergun's charge is "activated" (fired), it generates a bunch of plasma. Because this plasma was previously in a cool, solid state, it wants to expand really fast, and since it is confined by the barrel, this expansion is directed out the muzzle of the weapon. This is the same principle as when gunpowder explodes to propel the bullet in a conventional firearm, only in this case, the plasma itself is the "projectile".
I also understand that some do not believe that plasma weapons can fit in a handgun.
Conventionally, this is a reasonable attitude; you usually need a big honking power supply. Drake gets around this by postulating that the energy generation can happen via the charges, with the actual weapon being relatively simplistic. Again, see my analogy that the charges are essentially miniaturized nukes. In particular, they work on the same principles of one-time matter-to-energy conversion. Thus, a big power supply is not needed; all but the tiny initiation energy comes from the charge.
This isn't entirely unreasonable, as it is essentially how conventional modern firearms work; the gun is just a barrel and firing pin, while all the "interesting" stuff is built into the cartridge. (Okay, the ammunition feed mechanisms can be complicated also.)
It's worth noting that Drake never mentions how much energy goes into manufacturing a powergun charge. One way you could adapt Drake's basic idea is to invent whatever hand-wavium you like to say that charges are some sort of meta-stable arrangement of "canned energy". IOW, you feed a whole bunch of energy into some process which spits out something about the size of a bottlecap, which, when "jostled" in the appropriate manner, releases all that input energy very quickly; basically, a "hyper-capacitor".