My last two landlords asked me things like: “How long have you had this job?” “How long do you plan on staying here?” and “Are you looking to buy a house in the next few years”. I answered them all truthfully, then asked them the following: “How long have you owned this property?” “Do you have any plans to sell it in the next few years?” and “If I like living here are you willing to sign a 2 or 3 year lease?”. Knowing how long they’ve owned it should give you a decent idea on their equity status, albeit not a 100% guarantee.
Any landlord who is afraid to answer questions like that is probably hiding something, and I’d be verywary of signing with them. You don’t need to ask them specific financial questions about their private matters, but issues pertaining to your ability to remain in the house beyond 6 months are totally fair and reasonable.