Installed a new Cooper ShockSentry GFCI / DDFT / ICTA outlet in an outlet box that had no ground. White to silver and black to gold, without being grounded will this outlet trip correctly? It'll be handling close to a 10 amp load powering a large aquarium.

As the summer heat approaches, I am need of help figuring out what needs to be done to connect my a/c to cool the home. The a/c is a window mount type, and from what I can tell, has a 230/30A plug (see pictures). The home has an outside plug that is a 125/250V 50 Amp plug (see picture).. My question is how do I convert the a/c plug to work with the outside socket? I am most likely not going to do this myself, as I know very little about wiring... but I do want to know what needs to be done so that I can understand the process, and that I can insure whoever does do it, does it correctly. I am also wondering an approximate cost based on the number of hours it takes to do this. The circuit breaker box has a 125/250V 50 Amp breaker. Any help appreciated!!!! Note - The black socket is the 125/250 socket. The white plug is the a/c plug, the red sticker is from the a/c.