Advice on extensive damage

[Michael Nahm]

As I’ve received valuable advice from this site, I’m back for more Biggerpockets wisdom.

A rental property of ours had extensive water damage from a burst pipe that took place while the tenant was out of town.

Thankfully, the damage was discovered the same day it began, but the water affected the main level ruining the kitchen, bathroom, hardwood floors, carpets, drywall, and the finished basement which contained the furnace.

A remediation company practically gutted the damaged areas, and we have an estimate from a construction company, which is ready to go when given the go-ahead.    

I filed an insurance claim with my insurer, and they sent out an adjuster to the home.    
   
The adjuster than informed me that the home's HOA policy is the primary, and because of that, my insurer would cover the hefty deductible of the HOA's policy, less my deductible to them.

Since the home is in Minnesota, the insurer suspects the damage is due to frozen pipes. 
 
At the insurer’s request, we provided utility bills proving the home was reasonably heated.  

Now, over two months after the damage, and claim was filed, and apparently because they can’t deny the claim for not heating the home, the insurer hired an investigative firm a few days ago to see whether we neglected to care for the furnace (the furnace stopped working from the water damage, as the water reached the ceiling in the basement).  
 
I need to mention that the tenants are amazing, patient and responsible, and have been very understanding during this time, and their renter’s insurance is providing temporary accommodations for them.  

Another important piece of information is that the immediate neighbor of the home (it’s a townhome with common walls on either side) filed a claim with my liability insurance for some damage to their wall.  

My insurer denied the claim, as they explained it to me, because in order for liability insurance to kick in, either the tenant or me had to have done something wrong. My insurance rep further stated that since their investigation deemed that we had no fault as to the damage, they are denying the neighbor’s claim.
 
Please note that all this insurance jargon is new to me, as thankfully, I haven’t had to deal with these type of matters before.  
 
I’m wondering whether we should begin reconstruction while the insurer is taking their time, whether beginning repairs could affect the claim in any way, and whether what the insurer is doing is reasonable.