Can we capture and reproduce the bluetooth signals
Devices that can capture Bluetooth signals from devices they aren't communicating with/connected to are known as Bluetooth sniffers. These devices are most commonly used during development of Bluetooth devices to analyse the Bluetooth packets and protocols as they are transmitted. I have not come across one with APIs for replying packets over the air.

Most likely you will need a software defined radio (SDR) to be able to transmit arbitrary ("recorded") packets. If this is really something you wish to pursue I would recommend start by investigating what can be done by the very hacker friendly Ubertooth One: http://ubertooth.sourceforge.net/hardware/one/

Normal Bluetooth adapters, such as in laptops or the USB dongles you can buy, don't operate as Bluetooth sniffers and aren't able to capture traffic from the air. 

Bluetooth devices have a single IC for handling the Bluetooth RF reception and transmission, as well as at least the lower levels of the Bluetooth protocol stack. Some devices may have an external LNA (low noise amplifiers) for receiving and an external PA (power amplifier) for sending but these are increasingly being integrated onto the chip, meaning there are no separate components for RX and TX.