"I memorized a lot of kanji."
This sentence can be ambiguously solved as おぼえました and おぼえた mean the same thing but one is polite and one is casual.
You could say たくさん漢字を覚えました, but you'd never say "たくさんの漢字" to the best of my knowledge, because the 'of' in "a lot of (something)" is part of a fixed expression, not something that shows possession per se.
When it's used before the noun, you'd always need to include the の between them. It's a な-adjective (aka an "adjectival noun"), but たくさんな is rarely (if ever) used..
It accepted it for me, despite me mistakenly leaving out the の. Guess I should do a 'reverse reporting'. ^^
Yes, with the same meaning. Also, as written this means something closer to "(I) memorized kanji a lot". If you attach it with の like たくさんの漢字を覚えました it's more literally "a lot of kanji" rather than a lot of memorizing.