send btc from specific input without running bitcoind Is there any third-party API allowing to send Bitcoins using specific transactions as inputs?

Back in 2013 I had to run bitcoind myself and build a raw transaction manually. But maybe it's easier now, after 3 years? Running bitcoind is not so cheap now, but I really want to forward a bitcoin payment from one address to others, also keeping it safe and clear.
You can implement your own very light bitcoin client. It should support version and tx messages: https://en.bitcoin.it/wiki/Protocol_documentation

So, you make TCP connection to any bitcoin node with white IP. You can choose blockchain.info, for example. Your light client sends version message, remote node sends you getdata message, and you send back your transaction in tx message. After that remote node verifies it and spreads to other nodes, and to entire network.

You don't have to care about safety. Noone can steal your bitcoins if you correctly generated raw transaction.