We have had to upgrade our Integra DTC-9.8 (same generation as your 605) several times to fix issues related to HDMI handshakes, so even if the issues did not exhibit themselves in the past, it is still possible that your issues are related to out of date firmware.
HDMI cables can just all of a sudden become incompatible with a device. In our own projector configuration we have replaced several long cable runs because the cable all of a sudden just no longer accepts high bandwidth signals such as 1080p. This is why the first diagnostic we recommend to customers is to replace the HDMI cable between the player and the receiver; receiver and the display. Most handshaking errors are due to bad cables rather than defective hardware.
We generally recommend making sure that the receiver and the display are fully on and operating on their proper inputs, then switching on the player. It should not be required, but the handshakes can be more accurate when turning on the player absolutely last.
Pressing the button hard for an extended period of time can cause errors as there may be a loss of pressure, which will cause the player to think that a new command was issued. A strong, brief press should not produce these errors, however.
I hate to resurrect an old thread from the dead but I was wondering if the OP ever sent his unit in for repair of the sticking drawer problem. I had a BDP-83 that I sold to a friend and he is now experiencing this problem. Was curious about what was done and the cost of the repairs.