If a kernel update was required while adding PHP7 support it would require a restart, other than a kernel update then it wouldn't really affect anything.
I have a list of several
Recommended Hosts that I've worked with; you'd probably be able to get by using a Pro Hosting plan; think of it like shared hosting with additional resources and no webhost maintenance like a VPS would require. Both StableHost (my VPS provider) and BlueHost (Isaac's Pro Hosting provider) have these plans, regularly about $30/mo.