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Windmill Palms can be transplanted any time of year but they will have fewer problems if moved in spring or summer.
Dig out the new hole at least 1 to 2 feet larger then the root ball.
Plant the palm to the top of the root ball, important to not plant to deep.
Keep the palm soil damp but not to wet.

https://www.gardeningknowhow.com/ornamental/trees/windmill-palm/growing-windmill-palms.htm

Yes, almost any palm tree species can be transplanted successfully even when it's mature. This article explains how to do it:
http://edis.ifas.ufl.edu/ep001

I am not sure how big the Windmill Palm root ball is, but assuming 3 feet across by 3 feet down is a safe size for almost any palm. As explained in the article, the root balls for some palms are 2 to 3 feet, while others are significantly smaller.

Give your Windmill Palm as much sun as possible. If you don't have a good window location, you may need to use a grow light.

Brown leaf tips can be caused by sporadic watering, low humidity, build up of chemicals from tap water, and fertilizers.
It is a good idea to water palms with distilled water or even rain water. If necessary, let the tap water set out at least 48 hours before using.
You can flush the soil by running distilled water through the soil until it runs through the soil. Repeat this process several times and remove excess water from the drainage tray.

The soil should be moist from spring through summer, but barely moist in fall and winter.
Mist regularly while indoors for the winter.
A good average room temperature is 65 to 75 degrees F. Windmill Palm actually can tolerate a cooler indoor temp.

Fertilizer only in spring in summer. An application of a slow-release fertilizer, a rose fertilizer works well for this palm as it has the micro-nutrients needed.

ASPCA lists Windmill Palm as non toxic to cats and dogs.
I would advise checking with your vet.
If you are worried about human toxicity, I would call the Poison Control Dept. or check with your local hospital.

https://www.gardeningknowhow.com/ornamental/trees/windmill-palm/growing-windmill-palms.htm

You can do either one. Remove the stem close to the palm but avoid injury to the trunk.

https://www.gardeningknowhow.com/ornamental/trees/windmill-palm/growing-windmill-palms.htm

The shedding of the 'boot' happens at different ages of the palm.
This link has more information.
https://www.unce.unr.edu/publications/files/ho/2010/fs1072.pdf