(1.) What gauge wire is required from the converter to the battery? The original 'old style' battery was maybe a tractor battery.
The wire should be sized to match the output of the converter.
30 amps #10
40 amps #8
55 amps #6
70 amps #4
I'd aim for the smaller converters for this trailer. If the wire from the battery to the fuse panel is already large enough, wire the converter to the panel.
(2.) And should that be updated to a deep cycle marine battery? The old one is shot of course.
Yes... there's no point in using a starting battery. You don't need something really fancy in deep cycle batteries.
(3.) When wiring from the converter to the battery are there distance limitations? We have a 22' trailer. The original system is 12 v and 110 running parallel. The original battery box is on the front of the trailer and our plan location due to size, to locate the converter under the couch which is mid way curb side.
The sizes above are plenty adequate for an 12 foot run.
(4.) We are considering (if it would work) to take the 12 v from the converter and attaching to the trailer 12v electrical system.
The trailer electrical system is fused at the DC panel. Your converter needs to be connected to the input in the fuse panel so that no branch circuit sees more that it's rated ampacity. You can do that by connecting it to the battery (sharing the existing wire), or by connecting it directly to the fuse panel.
(5.) Would it work to 'back feed' to charge the battery from that location? Thus the wire gauge question.
Current typically flows to and fro in the lines to the batteries; this is normal. Remember that when the converter is on, current can flow into the battery and out to your lights, etc.
(6.) Should we consider 2 batteries?
Depends on how much and how long you camp disconnected.... and how
careful you are w/ lights, etc. LED lights are a big win here....
(7.) Is there a conflict to use a deep cycle marine battery and a tractor battery together?
It will work for a while; the tractor battery will fail first and
that won't help the other. In a pinch - sure - but it is definitely suboptimal.