And my goodness on the burglary!
Regarding your question, and I know this is difficult because his furniture was originally your moms, do you truly feel you'll use that furniture in the near future?
I'm asking because paying for a storage facility is expensive, and very often, what is put into storage ends up sitting in storage untouched for years and years.
So unless you have a move you're planning in your near future, you might consider selling or donating it. Remember, if you have photos of the furniture, you'll be able to save the memories without the need for space or storage fees.
That being said, if you do decide to keep it and it's going to be in storage for an undecided amount of time, I would opt for a climate-controlled storage facility. If you live somewhere where it gets very hot (or humid) or very cold, yes, there is a possibility that
the furniture will get damaged in time.
For everything else, I'm glad you're planning to order a dumpster for those items not worth saving. For those that are worth saving, but that you don't want yourself, consider an estate sale...and then donate whatever doesn't sell. Perhaps what money you earn from the estate sale could be put towards the storage facility fees.
A year later, I can see why you'd want this project to be finished. My thoughts are with you as you continue with this process. Hugs.