Show mice are like show dogs. Each standardized variety is bred to a standard set by a club. They are then shown against that standard in those of the same class. This is how it works in the FMBA at least. Other clubs, such as AFRMA (west coast US) or NMC (UK), function slightly differently.

Yes and no. There are varieties, some of which are standardized by clubs, but not all. These varieties often fall into various classes or sections depending on the club.

7 FMBA Sections: Self, Ticked, Tan & Fox, Pointed & Shaded, Marked, Patterned, Coated

7 AFRMA Varieties: Standard, Satin, Long Hair, Long Hair Satin, Frizzie, Frizzie Satin, Hairless
5 AFRMA Sections: Self, Tan & Fox, Marked, AOC, AOCP

NMC Sections: Selfs, Tans, Marked, Satins, AOV

All 3 clubs listed have some form of an "Unstandardized" category in which you can show mice that don't fit under other sections, but there may be some rules depending on the club.

Yes, but no. Breeders often breed a line for one specific variety, or occasionally two if they "mix" well. If you are breeding for cool dark blue self, you should not try to breed in reds or chocolates, that need rich warm pigment, or you will ruin both simultaneously.

An example of varieties that don't clash could be pink eyed white self, and ivory self. If you breed an albino and an ivory together, you will get more of both. (c/c x c/c-e)

Generally, most varieties need to be specifically focused on to achieve the best example of the standard. Dividing your focus can leave you with lack luster stock.

Pet store mice are generally mass produced to become food for other animals, with the slight possibility of becoming pets. Little attention is given to their appearance, if any. They simply have an entirely separate reason for having been created.

In the FMBA, conformation consists of: Condition (smooth coat, large size, overall health), Body, Head, Ears, Eyes, Tail, Temperament.

Show mice will likely have:
-Larger size, ears, tails
-Longer bodies and tails
-Lower set ears
-More striking coloration
-Crisp and balanced markings/patterns
-Better coats
-Calmer demeanor

Each variety has distinct traits it is bred for. Some varieties have more traits than others, and some traits are more important depending on the section.

Pink eyed white selfs that are albino, for example, do not need any attention given to pigmentation. They are naturally void of it. This allows the breeder to focus purely on conformation, and often means the variety will be "typier" than others. Type simply refers to the traits mentioned in conformation that have to do with the structure of the mouse.

In contrast, marked and coated mice are judged purely on their markings and coats. In the FMBA most varieties under these classes can even be shown in any color, not just standardized colors. They tend to have less type and structurally look closer to pet/feeder mice, but with a huge emphasis on the coat.

Selfs, ticked, tan & fox, and pointed & shaded tend to be typier than marked, patterned, and coated. Of course, not all lines will follow this exactly, and many will be somewhere in between.