Should the word market place have the m capitalized or not when used in the middle of a sentence?

only if it is the place that is called Market

If referring to the place/place where the weekly market takes place, market-place is one word

The answer for  both is no.

No. Only nouns and proper names must have capital letters. A market place is generalized and doesn't need to be capitalized.

No, unless 'Market Place' is the name of a product, or of a service, or of a public square in a city, etc.  

The single word 'marketplace' is known to my spell checker and to my dictionary, and is the form I have always known in Britain.   Do Americans write it as two words?

What makes you think that 'market place' or 'marketplace' should be capitalised?

No. It's not a specific title, just a generic term. It can be spelled as one word or two--I've seen it spelled both ways. Unless it's part of a title like Wilson's Market Place, it doesn't have to be capitalized. 

Los Angeles is considered a world market place for film.

Milan has some of the best marketplaces for designer fashions. 

Explore over 100 locally owned businesses that provide a unique market place experience.

The best market place for cars is in the city. 

"Marketplace," meaning the trading of goods or services for money, is a single word and is not capitalized. It is not a proper noun, the name of a specific thing, but a concept.

Examples:

This product suits the regional marketplace, but cannot compete in the global marketplace.

When it's two words, it's the literal place where a market is or has been.

Example:

Did you see that Trader Joe's has moved into the old market place?

"market place" are two words.