The details are a bit unclear, but here's some general guidance:
I need to make sure that their health insurance covers the nursing
home, but if not, what would happen legally? Can the nursing home take
my parents' house? Would all claims be deferred until after both
parents pass away, and then be made against the estate? Or what?
In general, the nursing home may (and will) invoice the cost to the patient, your mother. Now there are multiple possibilities:
- If your mother has enough money or other assets, she (or someone acting on her behalf) will have to pay. If need be, she will have to liquidate assets (such as a house, car, jewelry) to pay. There are some limitations, however, for example, a primary residence may be protected, and if the house is jointly owned, there may be protection as well, but it is not inconceivable your father may have to sell the house. That depends on the details.
- If your mother does not have sufficient assets, close relatives (such as the husband, parents or children) may be liable, under the rule that relatives are required to assist each other with respect to vital needs such as food, housing and medical care (under alimony and filial responsibility laws). However, again there are various rules and limitations in place, depending on specifics, and such laws do not exist in all jurisdictions.
Would all claims be deferred until after both parents pass away, and
then be made against the estate? Or what?
No, usually there is no such rule - that would be very problematic for any creditor, since you can't expect them to wait for payment for so long (possibly for decades). The claim will usually become due as normal, and it will be owed by either your mother or possibly her relatives, as explained above.
Someone told me that my father should legally remove my mother from
assets such as the house and bank account; otherwise the nursing home
would take them as payment.
That does not make a lot of sense. First, if she is (partial) owner of an asset, you cannot just "remove" her, as that would mean taking away her ownership. Even if she consents to giving away her property, there are usually rules allowing to reverse the transfer if it was only to avoid paying a creditor (often called "claw-back provisions"). So I would not do anything in that direction without legal advice.