"I draw a lot."
Iroiro is an adjective and only modifies nouns. It means various. Yoku is an adverb, here it is different from the adverb yoku which refers to frequency (meaning often). Yoi/ii is an adjective meaning good. To change it to an adverb that describes verbs you remove the final i and add ku. So yoku kike means to listen well. Kike is a rather rude command form though. You wouldn't use that in regular speech, unless you wanted to be really rude, I guess.
よく is an adverb. It describes the verb - ie. it tells us how the action is performed. In this sentence you have よく as the direct object of the verb as you have placed を after it and を always follows the direct object. Adverbs are not objects or subjects. They're just there to describe the verb. As for whether it is obvious that 絵 is what you're drawing - Japanese does not imply objects like that. Implied subjects/objects are almost always either 私 or あなた OR about something that people have been/are currently discussing - because everyone has been discussing it or are currently discussing it doesn't need to be repeated over and over again. Also 書きます means to write - the verb to draw is a homonym - different kanji.
I can't speak for the latter (though よく描きます。 was accepted for me), but for the former I think using が turns え into the subject of the sentence, not the object, so it might make the sentence into something like "The picture draws a lot." Someone more experienced please correct me if I'm wrong.
よく comes at the start of the sentence here because as a word describing the frequency of the action it is technically a time word and time words typically come at the start of Japanese sentences with the rare exception being you might change the typical/understood word order in order to emphasise some other aspect of the sentence.