"Will cheap ammo cause a gun to jam?" Probably not.
"My .22 Ruger jams if I fire it too fast." Rate of fire is not a factor in functioning. I have never seen a Ruger 10-22 jam with any kind of 22 Long Rifle ammunition when using the original Ruger magazines with steel feed lips. After-market, extended capacity, magazines with plastic lips (regardless of color) are prone to wear and damage, which can produce problems.
"I’m using Remington rounds." I doubt that your described problem is related to the ammunition.
Glad to see your on FR.