Also want fan1 to turn on at boot up and remain on
Usually, when people want a fan on all the time, it is easier to just use VCC at the terminals rather than deal with turning on the fan pins. But if you really want to use fan pins, you can put g-code in an auto start file on your SD card in your TFT, and it will be executed at the boot of your control board. This solution depends on you having the SD card in the slot when the electronics are booted.
fan2 to turn on after 5min when fan0(relay for router)is on and also turn off 5 minutes after fan0(relay for router) is off.
The 5-minute delays are a problem. Marlin g-code executes in a single sequence. You can add delays in the sequence, but nothing else will be done during the delay. If you put a pause at the beginning of the file, then none of your job will be executed until the pause is over. At the end, the job will not be completed until the pause is over.
To turn fan pins on/off, use M106 and M107 g-codes. I use fan2 on my Rambo board with M106 and M107 inserted at the beginning and end of my code to turn my router on/off.
In Repetier-Host, go to Config/Printer Settings/Scripts. Use the drop down to select when you want it to be run (Start Code and End Code probably), and insert the g-code you want in each section.
Most CAM tools have a similar ability to insert start and end g-code. I do it at CAM time so that I can run the g-code file off an SD card as well as a g-code sender like Repetier-Host.
As for turning your router/vacuum on/off, I go through some choices in this post. If you go with a mechanical relay solution, then I would use a 5V pin and M42 rather than fan pins and fan g-codes.
If you consider the delays important and are willing to put in some work to make it happen, it is possible to use an Arduino or similar and a relay module to make it happen. Your g-code would signal the start of the timers, and the control board would signal the Arduino to start timing and then to turn the fans on/off.