No, because -ul is not used with nouns but with adjectives. For example rossz (bad) > rosszul (badly). In case of magyarul the word magyar is an adjective, thus magyarul is as if we said *Hungarianly in English. (Of course, Hungarianly does not exist in English, I've invented this word only for illustration ....)
No, it is exactly like in English. The words magyar, angol, francia ... can be either nouns or adjectives the same way as the words Hungarian, English, French can be both nouns and adjectives in English. However, in case of magyarul tanulni the word magyar is perceived as an adjective (consequently magyarul is an adverb) while in case of magyart tanulni it is a noun.
A magyar nyelvet akarom tanulni. (Here magyar is an adjective, nyelvet is a noun in accusative)
Both are correct and as to the meaning they may be equivalent. However, it depends on the context which one is better (to say so). In Hungarian also the word order is important, more over the verb akarni can be conjugated in two ways (akarok and akarom) .... In other words, there are combinations that in English cannot be rendered quite exactly without a concrete context.
To complicate things even further
, the suffix -ul/-ül is sometimes used with nouns, too, but this use of the suffix is mostly restricted to literary or archaic language.
