"I bought a lot of clothes."
You could, and people do, just use 服 in this context. 洋服 is often used because the distinction still exists. However, "clothes" in English defaults to what Japanese people call "洋服". If you said "western clothes" to people in the U.S., most would probably imagine cowboys. So, unless the context calls for it, I'd avoid calling it "western clothes".
たくさん is a flexible word, it can function as a noun, an adverb, a の-adjective, as well as a な-adjective (but の is more common).
When you use it as an adverb it can basically go anywhere except the end of the sentence, and won't be followed by a particle. It would mean a lot of whatever action is taking place.
When you use it as an adjective you attach it to a noun with の or sometimes な and it modifies that noun, so a lot of whatever that thing is.
Not necessarily. The answer without the の is also accepted. From the previous comment, I suppose that without the の we might interpret the sentence as meaning (very) loosely "I bought clothes a lot", or "I did a lot of clothes' buying". As in, think of たくさん as an adverb modifying the whole rest of the sentence. Am I right?
To add to the discussion of 服 vs 洋服, it is more and more common for people to just say 服. Certainly if you just say 服 now in conversation, people will know you're just talking about clothes. 20 years ago the distinction was made more, but now most clothes are westernized, so it's like cell phone vs. smartphone. They're both still used, even though 携帯電話 is a little old fashion. But it's important to know both, or this case 和服 洋服 服.