Does the private key of Bitcoin change everytime the address changes? I have read from many sites that an address should change every time a Bitcoin transaction is made. Does the private key change along with the Bitcoin address. How is the address obtained from the public key? Is it the same method of deriving the public key from the private key?
If you are posting different questions I suggest, posting them up in different threads. But I'll try to answer all of them.

1) It doesn't work like that. Every address is a pair (private key, public key), the address is made after hashing the public key twice.

So, each address has one private key and private keys are not shared.

So if in one address you have 1 btc, and you want to send 0.5 btc to some other address, usually the wallet software creates a new address that receives the difference minus the tx fee, that's your change address and any further transaction that spends the funds will spend from that address (the change address).

To clarify, address A has 1 btc. You want to send 0.5 btc to some address B.

What happens is, the balance of address A is fully spent into 

B          (0.5)    &lt;--- amount you want to spend
C          (0.4999) &lt;--- amount you are left with after spending
fee        (0.0001) &lt;--- transaction fee on spending B
A (total)  (1)      &lt;--- original balance. fully spent


C is your change address, and when you spend your coins again you will spend from C and not from A.

2) The Public Key is obtained with one method related to how the way ECDSA works and the address is obtained by another method, mainly it is double hash of the public key that is encoded in a format called base58check as it includes a checksum to ensure that you didn't type the wrong address, also base 58 is designed to eliminate similarly looking characters, like 0 and O or l and I.

3) For the last question I'd like to refer you to How are bitcoins reassigned