For genetics 101, please read this page: Genetics Basics.
The answer to that is simply: mutations.
Truth is, our genomes are actually fairly error-prone. Every time our cells divide, the sequence must be replicated, but sometimes the ‘machinery’ involved make mistakes. Or, sometimes physical breaks in the DNA happen (through normal cellular processes or because of exposure to mutagens), and the broken DNA must be put back together, but it isn’t done perfectly.
Now, when these mutations happen in a cell, keep in mind it is just ONE cell. Just because a mutation happened in one, does not mean EVERY cell in the body is going to copy it and make the exact same mistake. So, what cell the mutation happens in is key. At perhaps the broadest classification, cells can be split into two types: somatic or germline. Somatic cells are all cells EXCEPT the gametes, which are germline cells. If a mutation happens in the somatic cells, it will not be transmitted to offspring. But, if the mutation happens in germline cells, then is will be transmitted to offspring.
So, to sum everything up, alleles arise when mutations happen in germline cells.
I’m afraid the answer to this is really unknown. Several recent studies have examined germline mutation rate in humans though. It is currently estimated to be about 60. That is to say, if you compare children to their parents, they will, on average, have about 60 mutations. For reference, the human genome is approximately 6 BILLION base pairs, and estimated to have ~20-50,000 genes. So 60 is not a lot, comparatively.
This doesn’t really answer how often actual alleles are created though, since many of these germline mutations are either synonymous (basically meaning it has no effect), or happen in non-coding regions of the genome. Some evidence suggests mutation rate may also vary on different chromosomes, or even in different regions of the same chromosome, too, resulting in some genes mutating more quickly than others. The takeaway I guess is, from one generation to the next, the chance of making a new allele is pretty low.
Because they are diploid, which means they have two copies of each chromosome. The same genes are present on the pairs of chromosomes, giving a horse two copies of each gene. But, by ‘copy’ I do not mean exact base-for-base matches. The alleles can differ, which is why horses can be heterozygous for genes. For a visual example, see below.
Fun fact: Not every species is diploid. Polyploidy (having more than 2 sets of chromosomes) is more commonly seen in plants, but does happen in animals (usually invertebrates such as flatworms). Polyploidy rarely happens in mammals, like the plains viscacha rat, which is tetraploid. For a genotype, they would have four alleles rather than two!
Remember how a horse has two copies of each gene? Well, these copies are randomly divided into that individual’s gametes. In simpler terms, each horse randomly gives their offspring one copy. When two horses are bred, the copy from the sire and dam are combined to make a diploid embryo. Check out the figure below for a visual example.