In a "factory" installation, your AC would be installed in a plastic pan that catches the condensate and channels it into the condensate drain tube that runs inside the wall. As far as I know, these drain pans are made to fit Dometic Penguin ACs, so one of several poblems that could be traced back to the installation could causing your leaks.
1) Your new AC might be installed in the original plastic condensate pan. The condensate drip holes on the AC might not line up with the channels on the pan, and therefore water is not being channeled down the drain properly, and instead is ending up overflowing the pan and dripping into your trailer.
2) These pans also have several holes drilled around the base of the exterior to allow water to drain out that has gotten inside due to rain. If some overzealous technician has sealed up these holes with caulk, then the pan will overflow, and the only place for it to go is inside your trailer.
3) In the even that there is no plastic pan, then it could be your AC is plumbed into the condensate drain in some other way. If the condensate drain tube is clogged up, then maybe the water is backing up and overflowing into your interior.
4) Your foam rubber seals that do all of the sealing of the AC to the roof, may not be tight enough, or they are in bad shape. Bad seals might explain the rain water getting in, but I think your rain water leakage and condensate leakage are related--see point 1 and 2 above.
So, step 1, figure out whether your AC sits in a condensate pan or not. It is hard to tell just by looking at it from the ground. You will need to get up close to the AC so you can tell if the base of the AC is sitting directly on the roof, or if it is in a plastic pan. The base of your AC should be metal, so if you see plastic, it might be the condensate pan. If you have a pan, then check to see if the holes along each side are clear, of if they have been sealed, or plugged by schmutz. Also inspect the condition of the pan. Sometimes a cracked pan can result in water leaking into the interior.
Step 2, you will have to remove your ceiling assembly so that you can look up into the guts of the AC. From here, you should be able to see where the condensate tube and any draining mechanism meet. Run the AC for a while and see if you can see where the water is leaking from. If the condensate tube isn't hooked up, then the problem is obvious. You should also be able to see the condition of the rubber seals that keep rain water out, and your cool air circulating inside the trailer. There should be three or more long bolts that clamp the AC to the roof. Check how tight they are. Also, confirm that there are spacers present in between the interior and exterior skins where the AC installs. If there aren't any, then you need to make some, as your AC isn't really clamping against anything but the interior sheet metal.