What tests are needed to see if an elderly parent should still live alone without any help?

You're right to get her the right level of care. Is there someone who can stay with her for 48 hours just to observe how she handles things on her own. I rarely trust the report from the senior as to what they can do, as it's often very off.

The place she's moving too may be ok for now but you may need to hire some in home care as she gets worse. Many facilities have a progression of housing and services, from independent living to memory care.

Mom loves to be around people and do things. Her short term memory is really bad at the moment--not sure if it's because Dad just passing on 11-5-15 or is it something more. I'm also concerned about her physical abilities to do things. She can move around OK, but her arthritis is pretty bad I think--she struggles opening jars, water bottles etc. I'm afraid she's going to need more care sooner than others in the family realize.

The test is: how is she doing now? Is she paying her bills? Is she taking her meds? Is there rotten food in the fridge? Are her clothes clean?
If she still craves social activity, she might better like a place with bus trips and activities. Even the shared dining is good mental stimulation and conversation.
A good senior apartment has pull cords or pendants for residents to summon help, with onsite response 24/7. Have you been to office visits with her and reviewed things with her MD?

It should be obvious if she is having memory issues and mobility issues. Her doc can run some simple tests for dementia but how's she doing? Falls? Short term memory? Leaves the stove on? Makes a mess in the bathrooms? Not to minimize the importance of testing but The deciding factor is her abilites, or lack of, not a doctors report.