While I am living in one side I am doing some renovation projects on the side I occupy: complete bathroom remodel, new tile floor (kitchen), new trim throughout unit, and painting/resurfacing kitchen cabinets. 

Replace rotted door trim around back exterior door, install a flashing on top of the back patio roof (seal the gap between the side of the house and the patio roof), level large stones in the back patio that have shifted over time, replace a couple pieces of broken vinyl siding, add transition pieces between vinyl floor and tile floor (2 transitions), fix a couple outlets that aren't working, replace cabinets, countertops, and add tile backsplash in the kitchen.This is obviously more than a couple days' work, but I'm capable of doing most of this myself, I'm just looking for some help to speed up the process so I can get it on the market faster. 

As @Theresa Harris mentioned, trim can be damaged by moving appliances in and out.  

Now that mechanicals are out of the way, put in drywall, time and paint, then trim carpentry.

A contractor took out the carpet and disinfected the contaminated areas and installed 6 fans to dry out the walls for three days. this costed $1900The next part is to install new carpet and install baseboard trims in the areas previously removed. they quoted me $3500 for that work.So total damage is around $5900Since this is my first time rental I need your advice on what to do.Should I directly bill the tenant that amount and ask them to pay it immediately?

We do the windows, get rid of dust and cobwebs, do touch-up, cut down/trim any bushes, etc. 

New roof, new flooring throughout, removed popcorn ceilings, 2 new bathrooms, new kitchen, replumbed, new lighting, all new doors and trim and a ton of paint.

I was trying to research online and it was mentioned that the neighbor that has incurred damage could have hired an arborist when he noticed a problem and have the roots trimmed before it led to more damage. 

You can see it in the pictures of our Buy and Hold Rental featured on my profile, it really makes the white trim pop and look fresh.

Using a vinyl product that is watertight in the field, I intend to first lay a thick bead of polyurethane exterior-grade caulk (Vulkem brand for instance) at the wall perimeter against the sub-floor and bottom plate and under the wood base-trim, and also at the door threshold, keeping water inside the bathroom.