Do I need an active internet connection at the airport to use the kiosk?

No. Kiosks will usually ask for your confirmation code for your flight (with your name) or the credit card you bought your ticket on. You don’t even need a phone for this if this is the path you’re taking. Note, some extreme budget airlines will charge you a fee to print your boarding pass on a kiosk at an airport, but none of the non-value airlines will do that to you.

My airline explains they can send a pdf copy of the boarding pass to my email address. Is it really not a problem to print my boarding pass on regular office paper and not on cardboard airlines use?

Not a problem to use regular office paper. I’ll frequently print on the back of a misprinted page just to save paper. Airport staff don’t care as long as it is legible and the barcode/QR code scans.

If I download the airline’s app to get the qr code to print the boarding card at the kiosk, will the airline spam me with ads I don’t want?

I’ll stay, “no they won’t spam you”, but that’s not fully exactly true. Some airlines will give you device notifications on airline sales or deals, but every single airline app I’ve seen lets you turn that off (but most are “on” by default when you start)

Note: you mentioned not having internet on your phone earlier. Many airline apps require internet to get your boarding pass. Again, its just a QR code with your name and seat, so you can bring up your boarding pass when you have internet, take a screenshot and have everything you need for your boarding pass without internet later.

Can I both print the boarding pass at home

Yes

AND get the qr code to print the boarding pass at the kiosk?

This QR code I’m assuming is your ticket purchase confirmation email/screenshot? If so, then yes to this one too.