Hi Rtrev, first off welcome to the forum glad to have you with us!!
As for your question about the benefits to having a dedicated IP, there are several.
(1) Your website is on its own server section, for better speed quality.
(2) if your running your own store front you must have this in place in order to be able to install a SSL certificate.
(3) The speed should be faster for your visitors and traffic in general, however I do not think it affects your search engine visibility. That happens based upon the content that is on your site, and your seo practices that you have implemented both on site and off site. Those effect the search engines more.
Also it does not have much to do with the emails, nor how they get delievered. However, typically you will have your own IP address rather than the servers shared Ip address when you do have the dedicated IP.It also can help to avoid more hack attempts, with a shared hosting plan your site is listed with many others together so if their site gets hacked many times it can also interfer with your files etc. Or at least that is what I was told by by hosting provider when my entire site got wiped out a few years back.
I hope my answer helps your question