In most cases there will be a monthly monitoring fee, though there are some options that do not have one.
Typically the customer buys the hardware but there are systems available where you can finance or rent the equipment without paying for all of it up front.
For most systems WIFI is required, though there are some older systems that run through the phone lines.
There are many DIY systems available but most comprehensive systems require professional installation, which is typically provided by the security company.
This will depend on the equipment you have and the security company you work with, this needs to be assessed on a case by case basis.