"I do not swim, but run."
Translation:Nem úszom, hanem futok.
Yes, you guessed it, "úszom" is the first person singular of an "-ik" verb. So, no object but still an "-m".
No - the correct indefinite first person conjugation of úszik is úszom (úszok is considered uneducated or hick).
Perhaps it is the Latin root for "fugit" (to flee) which is to be extracted from "futni", not the Germanic "foot".
It's likely neither - Hungarian is barely related to any European languages. Especially with verbs that have been around a while, there isn't likely any real connection, just false cognates.