What are the consequences for my dad who has Alzheimer's if I have the Dr. declare him incompetent?

Personal opinion..I am glad the guns are out of the house. Tools as well. (funny story about this I had a gate put on the deck so my husband could not get out, I put a latch on the other side so he could not see it so the gate was operable by others. My Husband took a screwdriver and began removing all the screws so he could get out)
Anyway concern yourself mostly with his accounts, make sure only you can get to them.
Try getting the VA to get the process moving as quickly as possible. Make the appointment with the VA doctor so that they can document that there is a need for Supervised care and that he can not remain in his house alone. Often the VA facilities have waiting lists so not sure how quick this would be. If he has other health issues that might make him eligible for Hospice that might speed the process along as they would probably make the next bed available for a Hospice patient.
As soon as you can change the locks on the house and get things cleared out as soon as possible. If the house is to be sold do that as well.
If you are both to inherit the contents and proceeds from the house you could in theory deduct the value of what your brother has taken. And if there are other siblings that would only be fair to them as well.
(inherit is after your dads death, until then everything in the house still belongs to your dad. As the person that is responsible for all his belongings you should do everything you can to keep them safe.)

My concern would not be guns and tools that Dad isn't going to use again. My concern would be his monies and his accounts. I would activate you POA, and then arrange to handle accounts, bills, etc and give Dad an allowance. Not easy, but could save his money from predators. If predators there be.

Im sorry about the loss of your mom.

Yes, I would activate your POA before brother takes dad to an attorney and has the POA switched to him. Unless that seems the appropriate thing to do? It’s difficult to be the Caregiver and not be POA. Is brother saying he will be caregiver now?
You may need to speak with your attorney about brother stripping assets but it is easy to let emotions run away at a time like this. Dad should not have access to guns.
Decide how your dad will he cared for and take action.
For the most part, being a POA is a thankless job but I’m sure your parents chose you for a reason. Let us know how it goes.