I've not heard of the need to wrap the antenna, so I can't answer that one.
As for what to do if you store outside but without a cover, much of its the same as with a cover:
- Sweep and wash the roof. Reapply Dicor or Alpha sealant to any seams or seals that looked cracked or in need of a touch-up. Make sure to check the sides of the trailer along with the roof. Dicor makes a LAP sealant that is non-leveling (for the sides) and one for self-leveling (the roof).
Things you can do without a cover includes covering the inside of your windows with dark paper to postpone fading of the curtains and sofa if it sits in direct sunlight for much of the day.
You can add an AC cover on the roof.