Can Tessel basically connect with any module found on SparkFun?
If yes, what kind of adapters/accessories would be typically required?
Generally, yes. The Tessel hardware API supports SPI, I2C, UART, and GPIO which covers the needs of most accessories. You may need to buy a 5v-3.3V level converter depending on the exact accessory you use (and I'm happy to help you figure that out).
Can any "Tessel module" be used with any other controller board running Linux and using the Tessel PTP?
The Tessel Modules aren't actually very different from sensors/actuators you would find on, for example, Sparkfun. They can be used with any microcontroller or Linux-based processor if it has the proper pins for communication (using SPI, I2C, UART or GPIO as described above).
I'm not certain about whether Tessel PTP is functional. It hasn't been worked on in a year or two. But you don't necessarily need it to use with other platforms. They will likely have their own API to the hardware communication protocols I mentioned above.