Is there any possibility to explore transaction chain via Original Bitcoin API (https://en.bitcoin.it/wiki/Original_Bitcoin_client/API_Calls_list) as the blockexplorer does it. If not, why?
You can get information on all transactions in the current chain with the "listsinceblock" RPC command. By not including a blockhash, all transactions will be returned.
From there, you can get further information on transactions with RPC calls like "gettxout", "gettransaction" and "listreceivedbyaddress" which can give you all the information that you typically see in blockchain explorers. But there's no user friendly way to view transactions, trace coins etc in Bitcoin Core. If you want something like that, you might check out open source blockchain explorers such as this: https://github.com/bitcoin-blockexplorer/old-blockexplorer-php