Let's say block 370,000 found and miners trying to solve next problem
Its a bit of a picky remark, but there is no "problem" solved durring block generation. Its more like drawing numbers out of a lottery pool and hoping for the win. Its just that no one knows which numbers will win until they tried them.
to create the 370,001 block but when the miners are trying to do this where the all that transactions stored ? local ? internet ? where ?
Unconfirmed transactions are stored in the miners (and other full nodes) memory (in RAM)
1. When you create a transaction it is send to other nodes (those running bitcoin core). Those nodes do not have to mine and most actually dont, they however verify that the transaction is sound and send it the nodes they are connected to. That way your transactions is communicated throughout the network very fast (almost instantly). A node might decide to forget about a transaction after a certain amount of time, but in the default configuration it does not do so, unless its shut down for some reason (e.g. someone closed bitcoin core).
1 edited after reply below