Miniature Antimatter Generators
I don't think this makes any sense. The problem is that current methods of creating antimatter require far more energy than you get from the matter/antimatter annihilation that results. If you have that much energy, why not just pack it into a bomb, or kinetic energy weapon?
First, to create antimatter, you squeeze a bunch of energy into one place. If done properly, the energy converts to mass according to $E=mc^2$. Half that mass is normal matter, the other half is antimatter. So the best case scenario here is that you put a bunch of energy in a bullet, use it to create a matter/antimatter pair, then recombine them to generate the same energy you already had in the bullet.
Second, you need a lot more energy than that. Of all the energy that goes into that tiny place to generate the particle/antiparticle pair, there's loads more that went somewhere else. The CERN website has some information here (it's related to the book Angels and Demons, but appears to be written by scientists at CERN, not the author of the book). It says that for every million collisions, you get 4 matter/antimatter pairs. That's about 500 thousand to one proton to antiproton. Further, each of those particles is energized with about 30 rest masses worth of energy. That is, we've dumped enough energy into a proton to create 30 protons if we converted all the energy to mass. So we're dumping about 15 million antimatter masses worth of energy into the creation of a single antiproton.
Third, there's even more energy used to do all of the above. There are magnetic coils, electricity to run the computers, etc. CERN estimates
About 1 billion times more energy is required to make antimatter than is finally contained in its mass.
Miniature Antimatter Batteries
You've got better luck here, but it's still going to be difficult.
The proposed methods of containing antimatter involve magnetic fields to keep the antimatter from interacting with normal matter until you want it to. I don't know how to calculate this, so I'll leave that for someone else to answer.
In general, however, I think you can make it about as small as you need. This paper on using antimatter pellets to start a nuclear explosion considers the possibility of a 1mm chamber to hold 1 µg of antimatter. The paper says
the levitation of a frozen $\overline{H}$ pellet within a 1 mm diameter cryostat at the heart of a complex thermonuclear device is a tremendous challenge for materials microtechnology. However, if metastable states of $\overline{p}$’s in Li−, Be− or possibly C−DT compounds are discovered, much simpler designs could be considered."
Since the antimatter reacts with normal matter, you'd have 2 µg worth of energy from the reaction, which gives around 180 MJ. From Wikipedia, an extremely large bullet (the 14.5 × 114 mm antimateriel round) has 32 kJ of kinetic energy. The .50 BMG round (still a very large round for a hand-held rifle) has about 15 kJ of kinetic energy. More typical rifle rounds have between 2 and 4 kJ kinetic energy.
This means a single microgram of antimatter could unleash about as much energy as 60,000 rifle rounds. Further, that energy would tend to explode outward from the contact point, so you're not just making a really clean hole through the guy, you're shredding/melting/plasmafying him to bits.
From here, they estimate 800 kJ energy released from a typical hand grenade (and point out it's not a safe assumption, so do your research). That means your bullet packs as much punch as about 200 hand grenades.
This Youtube video estimates about 11 kJ energy for a 1 $cm^2$ bit of grenade shrapnel. The M67 grenade has a diameter of 2.5 inches, so its surface area is about $4\pi(\frac{2.5}{2})^2$, or 20 $in^2$. This converts to 127 cm², giving 1.4 MJ energy in the hand grenade. Which means there are "only" 129 hand grenades in your antimatter pellet.
In general, you may want to scale the power down quite a bit to avoid killing friendlies and civilians and destroying the entire building. But I think your concept is workable here.
