Ganbare Goemon is "Goemon, persevere" or "Goemon, try your hardest" or something of the like because that's what gambaru means(to persevere, etc.)
By the way goemon isn't written 語えもん. If you use the IME, you get this first 五右衛門. I'm not sure which the game uses but you could probably look it up. Try
Google and search for ごえもん. Maybe try image search too.
The others I suspect are also wrong as 超(chou) means super and 執刀 means surgical operation but there are 3 other shittous I found. The 亜づえう is garbage from romaji not actually being japanese. You see Caduceus isn't japanese, something that really should be obvious. And for a tip, don't check for kanji after every character, put the whole word in.
Look these up first either with wwwjdic, jwpce, something because the answers are there unless you want more clarification like the form of gambaru or something like that.
Nou wo kitaeru otona no ds training I might ask for help with since I don't understand it really. Nou means braina dn kitaeru means drill, train, etc. So nou wo kitaeru means brain training/drilling but the otona (adult) and ds training confuse me. What meaning does that have?