You wont have to install windows apps in mint to make the machine work. And the backdoors are from software running in windows. Basically it was to do with the touchscreen driver. The device driver runs devices. They are built as DLLs which get run with administrator privilage so that any device driver can mess about with your system a lot because they need to. If HP wrote them then they get certified as being ok by Microsoft using whats known as WHQL a windows quality control and they send out a key so that the driver is signed by microsoft and then windows doesnt bitch about it when you try to install a driver that is signed by MS. What appears to have happened is that the drivers have a problem with them that allow remote execution - they are backdoored and it appears as tho this was not picked up in the certification.
So its no issue on mint since mint will run the linux kernel which has loads of driver built into it to run the touchscreen amongst other things. Because all linux is open source then you can look at the source code for any of the drivers so its a lot harder (neigh on impossible) to get backdoors fitted into common drivers and get them installed as default kernel drivers for linux. The kernel will also have a driver for the fan etc.
My opinion is to use Mint as your main OS and use windows to run the odd 2 apps that linux wont run - which is basically how I do it here.