If it's just a 3-4 page responsive site then just add on the domain to any one of your available hosting packages you already have. Using a VPS for a 3-4 page site is like killing a mosquito with a 12-guage. Add the domain to your Hostgator account and host it there. The shared IP won't hurt/help/affect your SEO at all. Neither will a dedicated IP, and there's no reason for a dedicated IP in this situation.
If you're looking for a new domain name, I've used -
http://justdropped.com for many, many years. It's a great (FREE) tool to help you find available domains using specific keywords. I've found great ones in the past with this tool.
Everything affects SEO but content is still king.
And don't use WordPress for a 3-4 page site. It's bloated and there is no need for all of the extensive code and database. Good luck.