Good rule of thumb for wood stoves is to keep everything connected to the stove at least 1' away from any combustible materials.
That old Ashley was a well regarded wood stove in it's day. I know a few folks that used them, mainly as a "parlor stove" but it served to heat the whole house. It has a full tin enclosure around the firebox so will be safer to use than a bare cast iron fire box.
I would disassemble the chimney cover and add in a screen with appropriate sized openings (not window screen).
The junction between the porch and the trailer is going to be troublesome to seal against water intrusion. I'd probably try a piece of aluminum formed to fit and sealed down to the trailer roof but I have a siding brake so could experiment. If you are friends with a siding installer, maybe enlist his help to make the junction piece.
I'm guessing this temporary shelter is in a warm part of the US as I see no insulation. If you need to heat this trailer, I'd suggest to get some good skirting built around it for the Winter. Skirting makes a big difference for comfortability.
