Could I leave this connected to the battery on my dump trailer while its attached to my truck?

yes, i leave mine hooked up on my camper, with 2 deep cell trogans. it is has a diode in it to stop back feed into the panel. 
 

Besides the obviously, what are the needs for the different wattage\'s?

My battery seems fine since using this. Higher wattage gives better ability to charge in adverse conditions. 
 

Anybody using this during overcast conditions? Wondering if it would be able to keep my boat batteries charged during rainy season.

Yes it will work fine during overcast. I purchased mine to keep my boat battery charged and it works like a champ. However during the winter when I store the boat I disconnect the battery and keep the charger disconected. I'll hook it all back up a day or two before the first time out for the season and it charges right up. 
 

Anyone know how to check output on this charger with a multi-meter?

Set meter to read dc voltage and touch leads to output connector. 
 

Can this be used on a 12v marine battery on a travel trailer?

yes 

Can this be mounted outside?

Yes it is sealed. However, when you use it in an area with heavy rain on many days you should protect it. If yes you can decide to keep it inside as well. I use one panel behind the (front) windshield and it worked fine. This way I can it use longer (many more years). 
 

is this truly weatherproof? i live at an apartment building and have no access to a outlet to plug one in.

Yes, it is.