Can I run my own software on my site?

This depends on a web host and a plan. Most plans will allow running scripts in languages such as Perl or PHP. Some plans will also allow you to compile program in C/C++ and run them. Some Unix plans will also allow you to run "cron" which enables you to automatically execute programs or scripts at a specific time and date. However to get a full control over all aspects of your server, you will need a dedicated or co-located server instead of a shared plan.