The rest miners checks whether the puzzle solved by the miner who claimed to solve is correct or not. Then how many miners approve to it; before it gets added to the block? Is it 51%?
No. The miner creates the block himself. There isn't actually a puzzle that the miner solves; he just does a specific mathematical formula (called a hash function) over and over again, slightly changing the inputs to that formula, until he gets a result from that formula (known as the hash) that is less than a specific target value. The target value is a value that is deterministically calculated from all previous blocks; i.e. given the same blockchain, you will always calculate the same target value for a given block.
There is no approval process or anything like that. What happens is that the miner will just broadcast the block to everyone, and everyone individually decides if they want to accept the block or not. This decision is based upon whether the block meets the consensus rules built into the software powering people's nodes. Other miners will also receive the block and decide whether they will accept the block or not. If they accept the block, then they will begin mining another block on top of it.
Blocks get added to the blockchain when nodes accept the block and add it to their local blockchain. There actually isn't anything that is "the blockchain", rather it is just that all nodes who are following the same consensus rules will accept and reject the same blocks so all nodes using those consensus rules will have the same blockchain.
Suppose a wrong entry gets there in the public ledger. Won't it affect the credibility of bitcoin? How it gets rectified? How a wrong transaction is rectified?
Blocks are not the "final say" of what is valid or not. If a block is invalid in any way (invalid block header, too large, contains an invalid transaction, etc.), everyone who is following the consensus rules that make the block invalid will reject that block and refuse to add it to their local blockchain. Thus those who are following the same consensus rules will still be using the same blockchain. Other miners will be doing the same thing and they will not mine more blocks on top of that invalid block. It isn't possible for a "wrong transaction" to be included in the blockchain because that would make the block that included it invalid to everyone using Bitcoin's consensus rules and thus the block is not part of Bitcoin's blockchain.
I also read somewhere that each block contains the information about previous block. So after 6 blocks is added to the block-chain; it is surely confirmed that the last 6th block is fully a valid transaction.
No. Blocks do not determine the validity of a transaction. If a block contains an invalid transaction or is invalid in any way, those nodes following the consensus rules that make the block invalid will reject the block under all circumstances. They will never accept the block even if the block has more blocks mined on top of it.
The miner gets the reward of 12.5 bitcoin. He adds this transaction to himself to the block?
The miner creates a transaction which pays himself 12.5 BTC plus transaction fees. The 12.5 BTC (the block subsidy) is created out of nothing; there is no issuer, there is no treasury.