How can I find out if someone is in the Schengen Information System (SIS)?

Under EU law, your mother has a right to inquire about any entry about her in the Schengen Information System. Because some entries can be sensitive (e.g. requests to arrest or even discreetly watch someone), the practical modalities are somewhat complex.

For France, you can read all the details in this document from the French data privacy authority (there is a template letter in annex 6). You will definitely need assistance from someone speaking French to formulate the request and read whatever they send you.

Note that it is quite unlikely that your mother would have been added to the SIS by France for a mere overstay noticed at the border. Other countries might do that but, as far as I know, in France it takes at least a decision by a judge (“interdiction du territoire français”), an “arrêté d'expulsion” or an “arrêté de reconduite à la frontière” (those are motivated decisions by a préfet, if your presence creates a specific risk – e.g. you are suspected of criminal activities – or possibly if you have been working illegally).

But that's not the only issue. Generally speaking, trying to reenter after an overstay (has she got a stamp about that in her passport?) is a bad idea. Border guards can always deny someone entry, even if no formal ban has been issued. They will evaluate her situation and if they think it's likely she will disregard the rules again, they should just send her back to the US.

And even without any ban or overstay, if she has stayed more than 90 days immediately before March 1, she cannot in any event reenter the Schengen area without a long-stay visa before the end of May. See How does the Schengen 90/180 rule work?

Finally, legally speaking, an entry in the SIS would make it impossible for your mother to enter for a regular visa-free short-stay but it does not block Spain from issuing a national long-stay visa, if they see fit. In particular, it's usually not an objection against visas issued for family reasons. So if your mother applied for a visa and the Spanish authorities tell her that some documents are missing, there is no reason to think they don't mean it. If she was banned, they could just say that. In all likelihood, her difficulties are not related to Schengen and the SIS in any way.

Beyond that, I don't know about Spain specifically but in some countries, the authorities just make life difficult for certain categories of applicants. You might need a lawyer to navigate the system.