Can a HP pavillion laptop run steam and play games?

What are the specs of your HP pavillion laptop?

If it has dedicated graphics card like these ones - http://www.notebookcheck.net/Computer-Games-on-Laptop-Graphic-Cards.13849.0.html then it will play games. Else no, if your laptop has Intel HD graphics card

Also, gaming laptops are generally cost $800 upwards. Here's high-end gaming laptop from ASUS with GTX 1070 graphics card
https://www.amazon.com/dp/B01K1INYD0//ref=as_li_ss_tl?ie=UTF8&linkCode=ll1&tag=basebal-20&linkId=265d97f6b6a8e47e5565786d45e38b61

Depends on the game(s) you want to play & the specs of that laptop

without a specific model number, I'd guess no

Yes it should run it. The performance depends on the specific model of the laptop.