Different pellets have different purposes so there is no BEST pellet. But I suggest that you stay away from cheap pellets they perform poorly. High quality pellets like H&N .177 8.18 grain Wadcutters, will cost you around $12 (USA)
Any air gun or rifle that uses a .177 Caliber pellet can use the Gamo .177 Red Fire Pellet. No matter who makes the air gun. The Caliber .177 is universal to all .177 caliber pellet guns or rifles.
If you are referring to casings like bullet casings after the bullet has been discharged, then the answer is "No." The entire Pellet is discharged from the gun when it is fired.
Use a .177 cleaning rod and push the pellet back out of the loading port.
No. Even if it would fit in the chamber, there is no firing pin on a pellet gun.
Cal .177
First, check to see that you have a BB gun, and not a pellet gun. Both pellets and BBs are made in .177 cal. A pellet gun will not shoot BBs (in most cases) If your gun IS a BB gun, you will need a gunsmith to check it for damage, stuck items in the barrel, etc.
Hammerli made 11 different Pellet gun models. Unfortunately you did not state the model number so here is the best I can do with out any information. See the link below
The best brand of .177 caliber gun that is good for hunting will depend upon what type of hunting a person is doing. Two good brands for general hunting are the Daisy and Crossman.
It can't use a regular cartridge, no.
Air rifles use compressed air or C02 to propel the BB or pellet. They do not use gunpowder. Their is no bullet shell and bullet just the BB or pellet.
The 620 is a 4.5mm or .177 caliber pellet rifle and is rated around 450 FPS.
I assume you are asking about air guns, if so then it would be helpful if you stated the model. Most air guns can accept all weights of pellet grains as long as the pellet is the correct caliber. In other words you can not use a .22 pellet in a .177 caliber pellet gun but you can use different weight pellets as long as they are .177 for that gun.