Laravel 5.1 has LTS which may or may not be of any importance to your project. 5.2 has some additional features not available in 5.1 that are pretty awesome. @JeffreyWay made two Laracasts series that, in addition to reading the Laravel release notes, should help you make a decision on which is a better fit for your project.
https://laravel.com/docs/master/releases https://laracasts.com/series/whats-new-in-laravel-5-1 https://laracasts.com/series/whats-new-in-laravel-5-2
As for hosting, Laravel is extremely flexible and runs well on almost anything that meets the server requirements
https://laravel.com/docs/master/installation#server-requirements
I have had projects on servers with several hosting companies but I can't say enough good about https://forge.laravel.com/ created by @TaylorOtwell and https://www.digitalocean.com/