tp53 example in Momand's book I am new to bioinformatics. While reading Momand's 'Concepts in Bioinformatics and Genomics' , I understood clearly that each codon translates to a specific amino acid.

But then, he talks about the TP53 gene, and then without explanation talks about P53... Is that a naming convention? It's not explicitly explained... Does that applies for any case? ie TP90 gene encodes protein P90?

I don't understand the naming: correct me if wrong, but they named the gene after the protein's weight, which is 53 kilodaltons? 
(It confuses me with the location of the gene).

Lastly I do not understand: if there is a mutation in the DNA, the mutation shall appear in only one cell I guess, how then the others cells' DNA start to mutate? or is it that all DNA in all cells are mutated instantly?
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.