IR = Infra Red (must be pointed to the receiver, low power consumption)
RF = Radio Frequency (no need to aim, may work through doors and even walls, high power consumption)
1). The air-mouse is an RF remote and to process RF signals the box needs to be up and running first. As a consequence you can power it off via RF but only an IR signal can wake it up again from standby. You can program the top power button with an IR signal to power it back on (just copy the power button from the standard IR remote to it). Being an universal remote this is not done at the factory for you.
2) To save the remotes battery the RF signal is paused when the remote is inactive. The first press just wakes it up again, the second press is a real button action. This is by design of the remote and not an Egreat/A10 fault. As an alternative you can turn it over to stop RF and again to turn it on again (works with mine, but I am not sure with all?).
3) Yes you can program the remote IR, but this is valid for the power button on top plus the 4 colored buttons only (an essential piece of information but indicated only correctly via the picture). Use the
TV button to toggle RF and IR output modes for the 5 programmable buttons. The pigeon English manual gives correct instructions how to IR program/clone these 5 buttons from any IR remote source.
Bottom line the Riitek MX3 (sold as multiple brands and even brand-less) in fact is a very nice and versatile universal RF /IR remote once its secrets have been revealed. No essential need for the original remote next to it once IR programmed for power on/off.

Use an all keys programmable alternative brand-less version myself. Use it with all my Android based media players by just moving the dongle. The Riitek i25 is a serious alternative having a different front panel layout and a via USB chargeable battery cell. Also back-lit equivalents for both do exist (disliked the buttons on those myself in practice).
