1. Ever since Bitcoin Inception and launched in 2009 till date, has it ever experience any form of burning?Not by the protocol, but coins have been intentionally burned and accidentally lost by users.
2. Considering the bearish and bullish run of bitcoin price, does currency burning has any effect on determining bitcoin price?In theory, burning coins reduces the supply so it generally causes the price to rise. There has been no demonstration of that effect however.
3. With the use of "eater address" how effective can it be in identifying currency sent to a wrong wallet address.Except for 1111111111111111111114oLvT2 (because its pubkey hash is 0), it is unlikely that many coins have been accidentally sent to any "burn" addresses. Here is a list of the known burn addresses with the most coins:
1CounterpartyXXXXXXXXXXXXXXXUWLpVr | 2130.83949557 |
1ChancecoinXXXXXXXXXXXXXXXXXZELUFD | 480.19421570 |
1111111111111111111114oLvT2 | 2.99181099 |
1BitcoinEaterAddressDontSendf59kuE | 1.91702244 |
1NewbiecoinXXXXXXXXXXXXXXXXDN67UA8 | 1.31817000 |
4. How can digital currency burn be carried out on a particular wallet address of a deceased person effectively without compromise. If someone has access to the deceased person's wallet or private keys, they can send the bitcoins to a burn address, but I don't know why they would. The idea of adding a consensus rule to burn (or recycle) coins that haven't been moved in a long time has been proposed many times, but its benefit is minor so I don't think it will ever happen.
On a related note, people are occasionally sent small amounts in an attempt to track or identify them ( so-called "dust" attack). You can safely get rid of those coins by creating a OP_RETURN transaction for each address in Electrum using the coins to pay the transaction fee.