If I remember correctly, you do not want to cross TX and RX.
Usually, you cross TX/RX because you are looking at it from the opposite side - the PC's TX is the Arduino's RX and vice-versa.
However, in the case of BlueTooth, the HC-06 has the same perspective as the Arduino - TX (transmit) is TX on the HC-06, since that is what the HC-06 transmits (over Bluetooth).
Also, if you are connect your Arduino via USB, the TX & RX are also connected via USB; I don't know if this would cause cross-talk. It shouldn't (so long as you don't send to your Arduino using Bluetooth AND USB at the same time), but it may. Also, as you're programming the Arduino using Bluetooth, you may get the Arduino's responses coming back to you via Bluetooth.
ADDITIONAL:
Go read http://arduino.cc/en/Hacking/Bootloader?from=Tutorial.Bootloader - this is all about the bootloader.
It seems from that that current bootloaders run at 19200, some older ones run at 9600. There are commands to program the speed on HC-06 - instructions can be found at http://www.instructables.com/id/AT-command-mode-of-HC-05-Bluetooth-module/ (this covers the hc-06 too).
Based on the bootloader tutorial, I think you have "less than a second" (only two specific boards are mentioned, so this is a best guess; I am amazed that it is this long, I expected a few milliseconds). If you have a USBISP, you could compile and upload a new firmware, with settings more to your liking - this is outside the scope of this article.
Another option is to create a completely separate uploader - an AtTiny could do it. Connect the AtTiny to the serial port, and then get the AtTiny to reset and program the board, or even program it via ICSP. This is a lot of work, and also outside the scope of this article.