Start with the original trilogy (Dragonflight → Dragonquest → The White Dragon). The Harper Hall trilogy (Dragonsong → Dragonsinger → Dragondrums) runs parallel and can be read alongside or after the original trilogy. Beyond that, publication order is generally safe for later novels.
No — despite being science fiction (Pern is a colony planet; dragons are bioengineered), the Pern books read as pure fantasy. The SF elements are background. The emotional core is always the bond between rider and dragon.
Yes — Todd McCaffrey has written several solo Pern novels (Dragonsblood, Dragonheart, etc.) and co-authored several with his mother before her death in 2011. These continue the universe but have a somewhat different tone from the original novels.