That....is a very tall order. It sounds like what you're asking for is some sort of "master algorithm to win battlecode" which, of course no player is going to give you. If they had such knowledge, they would use it for themselves
It's also not clear what you mean by solution. Solution for what? Do you mean like, for path finding? Combat? Search/exploration? I'm not really sure what you're asking for.
Also, individual Robots are limited by the scope of their own sight, so it would be very difficult to write any algorithm that can study all Robots, either friend or foe. Generally speaking, it's likely that no Robot will be able to see all of the other Robots at any given time. You would need to use all of your Robots collaboratively, and even then you likely still wont be able to see all of your opponents Robots.
As for your last point, it is indeed possible to write a distributed algorithm to do some heavy calculation/analysis using the broadcast system. It's not too much different from most functions. When writing your function, instead of using local variables, have it read/write variables from/to specific broadcast channels, in such a way that a new Robot can begin executing where a previous one left off. Be mindful of your Robot's bytecode limit, though. If your robot accidently runs over the Bytecode limit halfway through a loop iteration, it will stop executing immediately, possibly leaving itself or the broadcast array in a corrupt state. It may continue executing the algorithm on its next turn, only to find it was already completed by other bots, and error out unexpectedly.
When designing such algorithms, I find it best to write loops with all of the loop variables stored in the array, and at the end of each iteration, check bytecode levels. If approaching the end of my bots turn (less than 500-1000 remaining) I will write the variables to the array and terminate the function, letting the robot run its normal loop. If, on it's next turn, the calculation still isn't completed (or there is a new calculation) it can resume the function where it left off. Each robot in turn can do this.