Congrats on your purchase and expanding your portfolio!
-We need to set up an operating bank account for renovation work for our new property, should we set this up through the trust, through our existing LLC or just pay using our personal accounts? My father, my wife, and myself will all need to funnel money through this account to fund the project.
Operating account should be in LLC and you can transfer money and source it from there. Your living trust, if you've set one up, will notate that the LLC is part of the assets within that.
-We have been told about Umbrella insurance from friends. Should this be something we consider?
Yes, its additional protection for short money ($500-1000/yr) depending on coverage
-Will my father be able to transfer money into the LLC/Trust account without it being seen as a gift?
I believe trust is OK but LLC is more dicey - refer to your FA (Financial Advisor/CPA) for this
-When the time comes to pay back my father, will we be able to distribute proceeds from the sale/refi as we see fit?
You should be able to pay it from LLC and document as 'expenses' for the LLC. This is common practice.
Happy to share a CPA contact if you need it, just DM me.