I'm not an expert so don't bet your mortgage on it but here's an answer..
1.Why is this process(mining) made to be so difficult and time consuming?
If it wasn't so darn difficult, anybody would be able to re-calculate the tail of the blockchain and change transactions. If you spend a buck there's a "-1" in your wallet's history. With easy mining you could trace back the blockchain and recalculate with that same buck spent elsewhere. The absolute difficulty is not the thing though. It has to be difficult
enough to assume impossible for a single entity to have more than 49.999% of the total network's computing power. That should give you something to chew on.
2.if people stop mining there wont be any records(blocks) of new transactions..how does BTC solves this?
Not.
3.from what I understand a regular BTC client or BTC wallet doesn't do mining at all?
For what I know, the client doesn't mine. It only checks mined blocks. You can however use a command-line option to make it mine. Extremely slow.
4.Who verifies that the answer is correct?
All clients. A faulty solution is rejected, not added to the blockchain.