Does neder (or shavuah), which is given in my head (non-verbally) have any power? Also, is a one-time neder (If I won't finish this daf today I will fast tomorrow) is cancelled fully when cancelling it with a Rabbi after you broke it?

It's actually a machloket (=difference of opinion) based upon explicit p'sukim (=verses), where the Shulchan Aruch (Ch.M. 212) cites both psukim "מוצא שפתיך תשמור" ("You shall observe that which leaves your mouth") and also "כל נדיב לב עולות" ("and every generous-hearted burnt offerings) and in the end he holds that you can be lenient if you don't actually say it, and accordingly, that's what sefaradim do. On the other hand, the Rama (258, 3) decides like the second verse that even our heart's vow obligates us, and accordingly that what ashkenazim do. Regarding canceling it with a rabbi (Hatarat Nedarim), that helps for the future, but not regarding if you already "broke" the vow, which necessitates sincere teshuva.In general, King Shlomo (Kohelet 5, 4) already teaches us explicitly to stay away from vows, "It's better to not vow, rather than to vow and not fulfill", to just accept such commendable actions explicitly "bli-neder" (= "I'll do this without a vow"), for experience shows that life is full of unexpected surprises and forgetfulness.