1. What is the typical process of finding and hiring an artist? I posted a detailed description of what I needed on a site and said it would be paid and never even got one response. I was planning to pay through Paypal.
I actually go to artists, not public boards, most of the time. DeviantArt is a place to start, or conceptart.org, or even even here you'll find artists posting in the art threads, and you can look them up on their websites.
The idea is that you are not a top tier publisher, and nobody will line up to work with you, so you'll need to do some convincing.
Now, most freelancers are open to discussion once you actually initiate the dialogue. They might just not feel compelled to check on ads everyday, or may not feel like they are the perfect fit.
2. Is there some sort of contract that I should set up? I had planned to just make a one time payment for the assets. I do not want there to be any confusion where the artist thinks that they "own" part of the game and should get commission. They can own the assets, but not the game. Would just telling them the details be enough?
I recommend milestone or asset based payment. It helps reinforce trust on both ends: the artist gets paid regularly and knows you are not a scammer, and you get assets delivered on-time because the artist wants the money. Win-Win.
As for an actual contract, you can do so. I tend to work with NDAs just to set the tone, but it depends how high profile what you are working on is. To be safe, having a contract and/or lawyer on hand is advisable.
3. Finally, what should I expect to pay? I have no idea of what the price should be for these assets. I need one background image, a character image in four directions, and several accessories in each of these directions. I understand that there is not any set price for art, but is there any range which I should expect the price to fall into?
That is too vague. Level of quality/resolution is important. Content of the background, the style of art (pixel art? painting? illustration?) size and style of character, etc. all of that is more than relevant.
One way to do business is ask for a quote after providing technical requirements. Then, if the price is too high, adjust reqs and ask for a revised quote. There's nothing wrong with negotiating the price with the artist, quite on the contrary. If you build a strong understanding on both ends at the start of the engagement, both parties will enter that relationship with more assurance.
As for a range, I've paid assets anywhere between 0.25$-100$ and that's as an indie only. It can go much higher (especially with, say, full pieces such as concept art, models, etc.)