@Chris Wood welcome to the site.

With that said, it is a very old house and I am nervous about the potential for future problems/improvements (e.g. the windows are all wood framed single pane, the house is surrounded by trees resulting a lot of moss and debris).What are your guy's thoughts?

Blow them out before leaving the roomNotify landlord when away from unit for more than 3 days.Keep portable heaters at least one foot away from anything that can burn (including curtains, furniture, and you)No painting without approval from managementNo smoking insideAll furniture should have wood floor protection on feet. 

Then there was the fire in the stove (bought a new stove) and the dryer kicked the bucket (bought a used dryer) and other than that (knocking on lots of wood) the year was fairly uneventful.

@Jeffrey HoodI would do 50 year Architect shingles, remove all old shingles down to sheathing and replace and damaged wood also.

Is there any value in the wood?

This can be a HUGE plus, as it makes screening your own tenants a little easier, and it is usually the tenant who pays this fee.I have had a roof leak that resulted in completely destroying new wood floor in a condo that I flipped and was under contract at the time.

Consulting a site civil engineer would be invaluable in this case.I'm not well-versed on pest removal, but once you have the wall opened up and have removed any damaged insulation and wood, an exterminator should be able to help get rid of the roaches and geckos if you're not sure about doing it yourself. 

Which brings me to my question(s): does anyone know of a more cost-effective solution to common vinyl skirting, and is it necessary to build a wood frame around the trailer then install the skirting to it?

@Frankie Woods Thanks Frankie!