Tree roots and fence issue

[Samantha Serenes]

Hi, I need advice, please. Should I hire a lawyer or should I start fixing all this mess? Sorry for the lengthy story. 

The house I bought a year ago has a shared fence with a neighbor. At the end, there are 2 gates on both sides for my property and my neighbor's. I rent it to a tenant. It's in California.

1. A month ago HOA notified me that the gates didn't close properly and needed to be fixed soon by me. He thinks it's caused by my trees pushing the fence from the middle line. The neighbor has no trees. He notifies both me and the neighbor.

2. The neighbor said I need to remove all the trees (10 melaleuca trees) and replace the fence asap. The trees are too tall. Branches and leaves touch their windows on a windy day. And the trees are leaning against the fence. The fence is unstable. The fence is 20 years old by the way. 

I responded that I'll remove all the 10 trees and replace the fence at my own expense. The gate should fix itself once we have a new fence. 

3. The neighbor then added that the ground on their side is uneven. When removing the trees, they want me to lift up the brick paver on their side (they are small bricks and look easy to remove and put back) and check underground if there are roots and whether they cause foundation issues on their side. The fence is about 6 feet away from their house. 

I reported this to insurance and insurance said there is only 1 tree that touches the fence. The rest are about 4-6 inches from the fence. The roots are touching some part of the fence. Per insurance, the fence is still straight. 20 years aged fence contributes to wear and tear. They make it sounds like all these complaints are over-reacting. I'm still waiting for my insurance if they will cover at all or none. 

My preference is to remove all 10 trees and replace a new fence and be done with it even if the insurance doesn't cover this cost. But my neighbor seems to be a difficult person and now starts talking about liability for their tenant if they slip and fall due to uneven ground that they think is caused by my tree roots. I honestly don't know what causes the uneven ground. It could be the earth moving or the tree roots. 

My concern is replacing the fence and removing the trees may not be it. It could be opening a can of worms for more requests/ complaints. 

Do I need an attorney? Or should I proceed and do those 2 things? 

Thank you in advance for your time and for your help.