How do I control a resistive touchscreen?
Well you do measurement same way as on Arduino. The MCP3008 has 8 channels, so you can easily do 8 measurements. You'd normally connect two channels of the ADC to the X+ and Y+ leads. The connectors of the ADC are in high-impedance input mode, so you can treat them as disconnected as far as power supply is concerned. 

Then you also need to connect two GPIO pins of the Raspberry Pi to the X+ and Y+ connectors and connect the X- and Y- connectors to GND pins on the Raspberry Pi. 

When you want to do measurement for X axis, you'd set the Y+ pin to float, X+ pin to high and then use the ADC on the Y+ channel. Same way goes when measuring Y axis.

Now about the ADC: Well since ADC is analog to digital converter, it must have digital outputs. All ADCs have digital outputs, that's how the communicate with the rest of the system. The digital output pins are separate from analog input pins, so there won't be any communication problems. 

The particular ADC you linked to uses SPI. On Raspberry Pi, you can use hardware SPI port or use GPIO pins for software SPI. Do note that ADC requires SPI clock even when there is no data transfer going on in order to be able to run conversion.

Here are two relatively nice articles with a little bit of info about MCP3008 on Raspberry Pi: MCP3008 with hardware SPI, MCP3008 with software SPI.