Can a standard Rx/Tx bluetooth module send and receive at the same time?
The term you are looking for is “full-duplex”. 

“Bluetooth technology provides the effect of full duplex transmission through the use of a time-division duplex (TDD) scheme.”  

https://developer.bluetooth.org/TechnologyOverview/Pages/OverviewOfOperations.aspx

[Edit: since it gives you the effect of full-duplex you don't need a second antenna/module.]