You're right to be suspicious of things making it through the entire atmosphere unimpeded. Most of what Carl Anderson was detecting were the secondary showers of electrons and positrons produced when a high-energy proton (or sometimes another particle) from space interacted with matter. (It's not open-access, but here is the paper in which Anderson shows there is more cosmic ray activity up at Pike's Peak than down in Pasadena.)
1) How much antimatter is there to begin with? While I can't immediately find the number, I can tell you it's small. The vast majority of cosmic rays are normal protons, heavier nuclei, and electrons. Antimatter in space can come from high-energy phenomena like neutron stars, annihilation of dark matter of some sort, or as byproducts of very high-energy protons interacting with the interstellar medium. The third alternative is by far the most concretely understood and simulated; see for example the introduction to this Galprop paper. (Galprop is a widely used code for calculating the propagation of particles through the medium.)
2) The problem of interactions with air affects all charged particles we might want to detect on the ground. Electrons and positrons are discussed in the 1937 article "The Passage of Fast Electrons and the Theory of Cosmic Showers" by Bhabha and Heitler (can't seem to find a working link any more). Here, though, they mention a mechanism that does allow high-energy particles to make it through the atmosphere. When a single particle interacts, it might have so much kinetic energy that its byproducts are still energetic enough to cause further showers (and that energy need not be equally distributed amongst its byproducts). As they say, "The chance that an electron of $10^{12}$ e-volts should penetrate to sea-level and retain an energy $> 10^8$ e-volts is only about $10^{-5}$. We shall try to show in this paper that these difficulties are only apparent..."
