First of all, ofcourse you can beat the game without all the Aeons! It's just easier if you have them, and to win without them requires patience and careful planning. Whereas if you've got all the ultimate weapons and that, you just hit 'em with all you've got.
Well to get the cloudy/celestial mirror (which is what you need to obtain some of the ultimate weapon parts), you first of all go to the calm lands.
Take a chocobo (talk to the girl on a chocobo outside the albhed shop) and go to the cliff at the very south of the map where you first came into the calm lands.
You go all the way to the south-east corner, and there you'll see a yellow feather on the ground. Press (x) and your chocobo will automatically jump down. Now you go to across the bridge and turn right just outside the huge gates, don't enter the temple just yet. You can go back later. "Talk" to the Chocobo standing there. You have to beat the other chocobo to the middle without touching any posts. If you win (which is easy to do) you'll get the "Cloudy mirror". Your ultimate weapon gaining tool is now half complete.
Now go to the Macalania Forest, the part that you get to by entering it from the Thunder Plains. At the save sphere there, talk to the mother and her kid. Oops! Their dad is missing. Go right, right again (go past the path that leads to Bevelle)and then up the next path. There there will be a gathering of people. One of them is their dad. Tell him to go back to them, and then go back yourself. Well whaddaya know! The boy is missing now.
See the shiny path leading upwards to your left? Talk to the parents like quite a few times and head up the path. If a lady is standing there blocking your way, go back and talk to the parents some more. She should disappear.
Talk to the boy and he runs away. Now continue on until you see a big plant-flower-like-thing that looks kinda alive. Show the mirror there (press (x)) and it will turn into the celestial mirror.
This is also the place where you join the pieces for the ultimate weapons, so you'll be visiting it alot.
The rest of your questions someone else will have to answer for you.
