In cases like this I always like to check the board's schematic: Intel® Edison Boards and Compute Modules — Intel® Edison Breakout Board Schematic. The diagram below is the USB circuit of the Mini-Breakout board:
When the Edison is in Host mode, the MIC2039 IC is in charge of providing the power to the USB device attached. You can notice that the MIC2039 is powered from the "5V_SYS" line. That line is generated by the internal voltage regulator TPS62133 as you can see in the following diagram:
This means that in order to use the Edison in Host mode, it has to be powered from the J21 header (which requires an input voltage from 7V to 15V) so the voltage regulator can generate the 5V_SYS power line to power the MIC2039. Using the USB port in Host mode is not possible if the Edison is powered only with a LiPo battery through the J2 header.
At this point, what are the options available? Well, if you want to power the Edison with the system you described in your post (LiPo battery, Battery Charger and Wireless Charger) you would have to add some extra hardware. Basically you have to power the USB device using an external way. You have to use a boost converter to generate 5V from the 3.7V LiPo battery output. I believe you already thought in this option since you mentioned it in your post.
That would be my first option to try. I hope you find this suggestion helpful.
Thank you for the suggestion. I just realize that I can still use the same circuit just hooking up 2x 3.7 V batteries and power up the board via J21 Header.
Now I have one more question, can I power my Edison using the pin 4 on row J17 instead of using the J21 header? I took a look at the schematic and looks like it will work. I put a multimeter in the cable testing mode and it didn't "bip", but I thing it is because of the diode between them.
Yes, you can power the Edison through the J17-P4 input pin. Internally, this input is connected to the same line as the J21 header so it can accept a voltage input from 7V to 15V. You didn’t hear a "bip" with your multimeter because, as you said, there is a diode between them, but you can power the Edison through this input with no problem.