Hello, If i back up my wallet on bitcoin qt and I have other computer with bitcoin qt downloaded too. If I paste my wallet.dat file on my other computer and I do some transactions, will my balance also updates on the first computer too? Can I run the same wallet on multiple computer at the same time? Anyone know? thanks.
Yes. On both of your computer, when a transaction is made, it propagates among nodes. It will update the other client with the amount transaction. You can spend bitcoin anywhere as long as you have the private key. Private key is in backups, after downloading the full blockchain or if you use a lightweight wallet and restore the private key, your balance and transaction will update automatically. Bitcoin is basically about signing transactions, as long as you have the private key, you can sign and broadcast them, no matter how many wallets with that private key you run.