Corporations that produce roads maintenance

what does road maintenance mean. Will my roads get destroyed every months or something?

also if i have a factory that produces hospitals and i contract full production to my country does that mean that hospitals will be built every month? or does it just reduce the cost of building hospitals?

Road Maintenance is part of what your transportation budget goes to. Money is taken from your treasury and given to your stockpile in exchange for Road Maintenance, and periodically your stockpile makes purchases of Road Maintenance from the global market. That's when the money you spend on transportation every month actually leaves your country.

You won't lose roads if you run out of it, all it does is reduce your Supply Index by 4, and the stock of it goes into the negative. Since your Welfare Index is limited by your Supply Index, missing too many goods will reduce Welfare and your overall productivity with it.

I don't know about the second question. I think it results in the building of hospitals. AFAIK, they can't be stored in your country stockpile.

Yes it would result in the building of hospitals I believe, however I would strongly recommend against this because it's likely your corps will produce hospitals at high quality which will cost you more than ordering them at 120 quality. You usually want to build corps that use not to many low level workers and most importantly produce products that are in high demand.