Take a deep breath. Nothing must be decided right now. Every advisor will tell someone who has jsut dealt with a loss like your MIL did to not make ANY big decisions for 6 months minimum.
I assume she has a car - does she still have a license? Regardless she should take a few driving lessons through a professional driver training company. They would probably cost about $75 an hour - so 3 or 4 of those would be a terrific investment if it would get her back to driving and independence. She should get her license and driving skills back no matter waht is decided for her living arrangement.
As to where she should live - she shoudl stay there for now. She's alone - but is she ancient? Assuming she's reasonably healthy she should be jsut fine. Even if living in the house will require she spends some of her savings or life insurance proceeds she should still stay there for at least 6 months. If she makes a hasty decision now it could end up costing way more than it would to stay in the house for the next 6 months.
If she wants to come stay with you from time to time let her if it makes her feel better. It's temporary and not forever.
As for having her move in with you - ahhhhh! My mom, who I loved dearly, had an apartment on the side of our house. She had her own living room, kitchen bathroom, washer & dryer, entrance etc. BUT - we still got on eachother's nerve. It had many, many very positive aspects - she was here for the kids when they were little, we were here for her as she aged and came home to recover from various surgeries, etc. (she had heart surgery, cancern surgery, hip replacements,etc.) My mom passed away a few months ago and in retrospect I am so very glad that I had the opportunity to be so close to her in the "Autumn & Winter" of her life. I can't imagine not having been here for her and my kids have so many warm and tender memories that their cousins don't have becuase we lived under the same roof. BUT - it represented many challenges over the years and her presence did impinge into our life. Overall there were more positives than negatives - but it was very difficult at times.
If you eventually decide to have youru MIL move in you must build her her own space, kitchen, wahser dryer. I have to say I don't think a basement apartment is what she'd want unless you live in a high ranch so the downstairs is ground level. Older people begin to have arthritis and other maladies that willl be made worse by the dampness of living in a basement - and very little daylight! Do you have a garage that you can make into a studio apartment? (We built a huge shed and mom's apartment was built into our 2 car garage) If it's only a one car garage can you expand it into your back yard?
If MIL sells her house and pays to have her apartment built it will also increase the value of your home and you won't be upside down on yoru mortgage anymore...
As for selling your current house - i would never do that. The bank won't jsut take a loss. Even if they are willing to take a loss on some of it they will still want you to take on some of the balance and it will damage your credit rating.
I understand how it all seems overwhelming right now - but as you have told your husband, it's ultiamtely your MIL's decision and assuming your husband has siblings there are other voices she will listen to. Help your MIL organzie her decisions. Right now she's in shock and still grieving - which takes a long time. Every major decision should be evaluated with a pros / cons listing. But I cannot stress enough that she should not sell the house for at least 6 months. (Realtors will swoop in on a recent widow like vultures!) Whatever the cost for her to stay in the house for 6 months will be less than the cost of a bad decision. 6 months from now if she still wants to sell the house then she'll be less emotional about it and feel certain about the decision.
IN the meantime encourage her to connect with her friends, chruch, senior center, etc. Also, BTW - many townships, counties and other local governement have senior transport available. There may be a few forms to fill out - but it might be worth it for MIL to get a ride to the grocry sotre once a week or to the library, etc.
This is just a season in your life that will be busy. It won't last forever. I spent alot of the last few years as the primary caregive for my mom, bill payer, financial consultant, medical transporter, the one who "made her move to a nursing home" etc. and it seemed overwhelming at the time - but it was really something of a blessing and honor to be the one to accompany her on the journey.
You're a good DIL.