Your Pokemon games or your game system might be broken. A couple of my friends had that problem.
Most likely, because your game's internal battery has run out (though you didn't specify what the exact problem is, e.g. it doesn't turn on). This tends to happen to old cartridge-based Pokemon games.
Action replay is a software with pre-loaded cheat codes that work across many games including pokemon white
Yes you can trade from past Pokemon games...I think only heart gold/soul silver though...
No. How can you catch a Pokemon if you cant weaken it. In Pokemon ranger though you can capture your partner Pokemon.
No. Pokemon Black and White have new Pokemon that the past games' engines cannot recognize. Even if you battle with Pokemon from previous generations, they might be loaded with moves from the new generation.
The internal battery is a battery in the game pack that stores energy to keep track of things like the passage of time when the game is not in use.
No. Pokemon Black and White are the only current games with no legendaries not belonging to their region. It can be traded from other games, though.
swarming Pokemon are Pokemon that appear daily on a certain route. you have to have obtained the national dex first though on the games that they appear on.
No, eggs are only available in the generation one games, there are by breeding Pokemon in two and up though.
No Pokemon does NOT exist and it won't because it's only a cartoon. It does exist in games though as well as tv
Because the battery is dead in the game preventing the game from saving.
All Gameboy Games can save if the battery is working. If the game stops saving, you may need to change the battery.