This is possibly more of a molecular genetics question appropriate to biology.stackexchange.com, also note that it's usually better on stackexchange sites to restrict your post to a single question and do some preliminary research via google or your search engine of choice. But I don't see any harm in answering it here nonetheless.
The real answer is that naming conventions for genes/proteins are incredibly inconsistent and counterintuitive. You can't really draw any conclusions about a gene/protein based on its name alone. For example, some proteins such as p53 are named for biochemical features, others are named for phenotypes observed when their gene is mutated, such as for instance the gene decapentaplegic in fruit flies. Often they involve little jokes, such as the gene called mothers against decapentaplegic which represses the decapentaplegic mutant phenotype. So it's all a bit of a mess and people kind of assume that you'll follow along, which can be confusing.
It's a little rarer that gene names differ substantially from the names of the proteins they encode, but not unheard-of. This is indeed the case with your TP53/p53 example. See a short explainer here or a longer one here. What you'll notice is that there are in fact multiple different names for the same gene/protein. What you'll often see is that the shorter name (p53) is the one that people tend to use colloquially.
For your second question regarding how mutations are propagated, the short answer is that mutations are inherited from mother to daughter cells. Cells don't generally "infect" other cells with mutations (though in biology there are always exceptions). I think that what you are interested in is the difference between "somatic" and "germ-line" mutation.
Hope that helps.