So i can check what services are running on my computer (or on the instance i connect to in SQL server manager) with the following statement
EXEC xp_servicecontrol N'querystate',N'Service Name here'
Which will return either Running or Stopped
How (if any) can i check remotely for services that are running on a server on my network that does not have SQL management studio on?
In short using MSSQL to monitor services is not going to be the most successfull way of monitoring services. There are plenty of monitoring solutions out there that will do this for you and also alert you when things go wrong.
See the following answer for some information: Monitoring uptime and health of several services (web service, oracle, web app)
The solutions mentioned in this answer are all open source and free and I have personally used some of them on servers where we had over 30 million registered users and 350 000 concurrent user connection. Without these monitoring solutions in place my live would have been hell. They work out of the box and allow you to monitor a whole range of things.
Leave SQL server to do the database work that is what it is for.