1) You know you've received Bitcoins if your balance in your wallet increases.  Any Bitcoin wallet software will indicate your balance and provide you with a history of your transaction record.  I'm not really sure what else you meant by this question.  If you check your transaction record, you will know if you received (or sent) bitcoins or not.  It's really not any different than looking at your bank statements to know if and when you've received a deposit.

2) Miners act as a lubricant for the Bitcoin payment network.  Miners are required to confirm and process transactions.  If you remove all miners, you completely halt the network.  That is why miners receive a subsidy (i.e. block rewards) for their efforts; the subsidy provides an incentive for miners to keep mining, and this ensures that the network is able to continue processing transactions.