It sounds like you are in a very stressful situation.
Legally, the question of wheher Dad can sell his house would depend on if he has capacity. The fact that he is having conversations on the phone with you about it, would suggest tnat he does.
However as you pointed out you are the only person with the keys and practical matters would be beyond him I suspect.
Coronavirus restrictions on selling a house are lifted now but practically you are not yet able to help him. It sounds like he is anxious about it but at least it doesnt sound like he is seriuosly trying to go back there, which a lot of people in care homes are always asking to go home.
You have PoA so legally in his best interests you can istruct a solicitor, engage a house clearance service, if you need to you can also engage a professional cleaning service, decluttering service, whatever you need to delegate hands on to help you get it done. However if you are sheilding or self isolating for medical reasons you may just have to wait intil the right time.
The LPA gives you poeer to make decisions and arrange things on his behalf. What it does not do is act as a legal obligation enforcing you to do anything yo are not able. If you are unwell do you think it would be wise to stop acting as the attorney, this is possible if you felt you are no longer able. I'm including a link to the official gov. uk guidance
Acting as an attorney - duties, including registering a lasting power, starting to act, gifting, handling disputes and replacement attorney responsibilities.
