Since you're new to affiliate marketing, I would suggest you avoid PPC to start with (there are many free resources available on this forum and elsewhere that discuss PPC, but for now avoid PPC as it will eat up your budget). To answer your question, yes, you can promote a product without a website, but having one is far more effective. In the past, I promoted products on craigslist (be careful if you use this method, and try not to come off as too "salesy" or "spammy"). Another method not mentioned is to do video promotions on youtube.
You can try social media, PPC, or email traffic. All require just as much, if not more, effort then creating a web site though. There's no 'fast way' to make cash that's sustainable. Set yourself up for the long haul now, so that you make money for the present and future.
You can but you would have to check if it's okay with the affiliate network because not everyone allows it. But I suggest to create a blog, you can outsource content to fill up your blog.