I understand this problem… I have thought myself into a corner on this one more than once. :o
What I have finally come to understand is that while God is eternal, and while there does not seem to be any reason for our prayers to be effective means of “making something happen/asking for some result”, there is something else to consider - our free will.
God created us in His image - throughout the ages, man has understood this to mean that God created us with free will; with reason and a capability of making decisions using our own intellect. This is what separates us from the animals. (The whole angels/demons/humans thing is a whole 'nother topic… maybe next time :))
As someone else said here, by our prayers we are participating in the Body of Christ, with Jesus as our head. Our prayers are effectual because we freely choose to give them, and because of the merits of Christ.
It doesn’t really matter, for our purposes, that God is eternal (as you acknowledge) and therefore that all times and choices are known to Him – just because He knows them doesn’t mean that He has in some way denied us the freedom to make them. Apples and oranges.
Now, my next point is - don’t fall into the trap of the philosophical “either/or”. (ie, “either I have to pray for this person or he will be condemned to hell”) Christianity, and Catholicism in particular, has very few “either/or” components – this, like so many other things, should be thought of as a “both/and” proposition.
Think of it this way - while God has promised that our prayers (in this case, prayer defined as our decision out of our own free will to ask Him for something) will be heard and have some kind of effect (even if the answer is “no” or silence… it is still heard and is effectual in some way, whether or not we can perceive it), God Himself is not limited by prayer! Thus, while our prayers are helpful, because our free will allows us to participate and cooperate with God’s grace, God can still “save” someone without our prayers if He so chooses.
So - what we learn from this is that God’s ways are not our ways. We know that prayer is effective, God has told us so, and there is even empirical evidence that this is true. We also hold that God knows everything, including what decisions we will make. It seems to me that what can be extrapolated from this, is that God knows when prayers will not be made – and Himself wills whether or not to do something without a prayer for it (again, don’t get bogged down into the earthly “help” either, eternal salvation is what’s really at stake here).
Bottom line - prayer is important, but do not become overly anxious over “I can’t pray for everyone” (you can though, just say, “Lord, I pray for everyone who needs prayers” :)) because, as you said, God is Mercy and Goodness (God is also Justice, never forget that), and we will have to trust that if we do our best to fulfill our vocation on Earth, He will take care of the rest. God’s mind and reality are so far beyond ours that there can never be any comparison, and there is (in the end) no way for us to truly grasp what is happening on Earth. I pray that someday in heaven (praying that I get there of course!) I will be able to understand this more clearly!
Hope this helps at least a little… it’s really late and I’m really tired