You can't take the processor off the Teensy board. The chip which makes the MK20D work like an Arduino is not available unless you buy the whole Teensy.
Yes, definitely space the Teensy up or put it on the back side so that components can go inside its footprint. I would not expect to put more than one or two components on the back side unless you are really slammed for space.
If you get a Teensy 3.5 or 3.6 then the SD card reader is included for free. It's the same width as the other Teensies but obviously a bit longer.
For a motor controller, the VNH7100 can handle up to 18A max and it's tiny. It's also very low RdsON so it won't heat up much. I've never used one before but I just got a bunch in the mail to try out for my next design. At 100milli-ohms it's not going to generate a lot of heat.
For switching general-purpose devices like a fan - including PWM control - I would go for a chip like the BTS716G. 4 outputs with RdsON of 140 milli-ohms. For high current, you can pair up channels or even all 4. BTS5012 is single-output but it handles more amps and has current feedback so you can see if your fan is drawing power.
For the audio amplifier, buying that Adafruit item as a pre-built module is going to save you a gigantic amount of time and effort. If you can mount that over the top of your 4"x2" board then it will be silly to try to copy that into your main PCB.