I haven't had such good luck with the 177 for killing power. I used to use a 177 RWS 34 spring gun, but once I got into 22 pellets I have never looked back.
As far as minimum pumps to effectiveness, that is an excellent question and I think there ae more variables than we can address with a simple "This many" response. First, the gun. How much punch per pump you get will varry dramaticly from one make to another. For pumpers, I like the Benjamin Sheridans. Well made in America. Then of course there is the pellets. Weight; flat, round or point? And of course distance will matter. All this before addressing placement and target species.
OK, so now what, you may ask. I usually evaluate a new airgun's effectiveness as soon as I get it. Some are only good for targets, others can drop vermin. I start with the air gun and intended pellet, then try accuracy at various distances. Then penetration test. From empty soda cans, to empty soup cans. If it doesn't penetrate the soda can, it won't kill anything. Will it punch through one side or both? How about the soup can, a dent? A hole? You get the idea. If you do this in your back yard, you will find out if the neighbors are going to call the police over some BB gunning. That's good to know. You know better than I do if that's something you want to try in your neighborhood.
In your case, you will want to try a 'controll' target and then try it with different numbers of pumps. Maybe an accomplice to report on how loud it is from a given distance.
Since you were disapointed with your last pump air gun, by all means upgrade. Like I said, I like the Benjamins. There is a ton of new stuff out there that I have no knowledge of. Others may be better able to give you some input on newer options. The Benjamins are classic, and there is some to be said in favor of that. The 22 or the 20 will probably serve you better than the 177. Maybe a scope?
Good luck and good hunting.