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Yes there's a cartridge puller made for just this purpose. You can get one at any plumbing supply house and maybe even the big box stores
Probably, are you sure the inside of the valve was clean before you installed the new cartridge? Often neoprene or other residue is left behind causing problems. The only other likely possibility is using the wrong replacement cartridge
Could also be a bad cartridge. I never use the plastic ones.
Try a plumbing supply house for a brass cartridge