No, the French really don't care if your husband got deported from the US. It's virtually unheard of for two agencies within the same country to share data like that. Even less chance that the different countries are going to co-operate like that anytime soon.

As far as getting visas for yourselves, you need to check the website for the French consulate in Mexico for their specific requirements. There is nothing stopping you from buying property in France, but owning property won't give you any sort of advantages as far as getting a long-stay visa.

Basically, for a visa, you need to have a job lined up and the employer to have filed for and obtained the work permission for you in that job. (Has to prove that there is no local or other EU national available.)

As far as setting up or investing in a company in France, it would be a whole lot easier if you were already legally living in France. If you have a particularly innovative idea for a business, you could try for one of those "Skills and Talents" visas - for which you need to develop a project designed to benefit France, your home country or relations between the two.

Your case for a visa (or visas) would really be helped if both of you were relatively fluent in French. (It would also help immensely in the job hunt!) It would also help if you both have some ties to France - friends, family or frequent travel there. Offering to go back to Mexico on a regular basis could backfire, as it suggests that you aren't really "committed" to the move - and it does mean that you won't be dependent on your French income and thus may or may not be intending to pay your share of taxes and cotisations.

Your best bet at this point might be to make a couple trips over to France to start scouting out options - jobs (see what might be available to someone with your qualifications), places to live, whether or not you can cope with the culture, etc.
Cheers,
Bev