1) Yes, you are right. This may be because MBR3 is in sleep mode.
You could set the Scan period in CapSense sensor configuration tab of Ez-Click to 20 ms. (This is the scan rate in look for touch touch mode)
You could increase the state timeout further, to prolong the active state.
The controller will not go to deep sleep mode, unless a sleep command is issued. You could refer to section 7.1 you have mentioned earlier.
2) Yes, changing debounce to 1 and disabling IIR and median filters should help.
Please try the above and update the response time behaviour.