Any tips on re-carpeting doghouse?

I've put carpet in vehicles. A lot of it can be cut or formed as you go but templates are like gold, just allow for extra.

If the carpet is jute backed, a spray down of water will loosen it so forming is a snap.
If the carpet is plastic backed or has a pad attached, heat can help but pressure and fastening while the adhesive sets is more typical. You can also fold or roll some plastic-backed carpet to loosen it before installation. All carpet has a weave and weft (backing is laid in two opposing directions) so work it in two directions.
If you have a carpet with dense pad, you may have to make some cuts in the pad to form around tight radiuses (radii ?).

Good luck with the installation!

How you carpet your doghouse is kind of a personal thing. German Shepherds generally prefer utilitarian carpets, being they are working dogs. Afghans obviously prefer Oriental Rugs. And a Shih Zhu would probably need a toilet paper motif.

You wont be able to form the new carpet over the doghouse be heating it. The best way is once you have the carpet removed use it as a template to cut the new carpet. Carpet can't be formed over compound curved sections only single curved or wrap sections. The original installers know this so they made the cuts in the right places.

Thick trunk liner should work.
https://www.amazon.com/Automotive-Trunk-Carpet-Medium-Speckled/dp/B006320UKY

Use a heat gun to loosen the old carpet. you don't want to gouge the fiberglass dog house with a saw.
As to the new carpet, lots of cutting and trimming to make it fit.