Je schneller wir laufen, desto wärmer werden unsere Muskeln.

The je…desto (or je…umso) construction creates a correlative comparative meaning “the more/faster…, the more/warmer….” It always follows this pattern:
• Je + comparative adjective/adverb + subordinate clause (verb at the end)
• Comma
• Desto + comparative adjective/adverb + main clause (verb in second position)
In your sentence:
• Je schneller wir laufen (the faster we run)
• desto wärmer werden unsere Muskeln (the warmer our muscles get)

Comparatives in German usually take an –er ending:
• schnell → schneller
• warm → wärmer
Monosyllabic adjectives with a single vowel often get an umlaut in the comparative: a → ä, o → ö, u → ü.

The first clause after je is a subordinate clause, so German word‐order demands the finite verb at the end:
• je … wir laufen
The second clause is a main clause, so it follows the normal verb‐second rule:
• desto wärmer (adjective) werden (verb) unsere Muskeln (subject)

Werden + adjective expresses a change of state (“to become” or “to get”).
• Unsere Muskeln werden wärmer = “Our muscles are becoming/getting warmer.”
Using sein (to be) would simply describe a static state (“Our muscles are warm”), not the process of warming up.

• Muskeln is naturally plural here (we talk about all muscles).
• It’s in the nominative plural because unsere Muskeln is the subject of the main clause:
desto wärmer werden unsere Muskeln

Absolutely. Umso is interchangeable with desto in this construction:
• Je schneller wir laufen, umso wärmer werden unsere Muskeln.