Using the usb port for lcd display

If i use a 5v power supply instead of the usb port,
can it be used for controlling lcd display?
something like this:

and another question, can this display work with 9v battery?

Assuming you mean the USB port on the April board, no; it is for power only. If you want to use that LCD from adafruit and a 9v battery then you will need a logic level converter to convert the ~3v Imp logic level to the 5v needed for the LCD module. You also need some way to make 5v (e.g. a linear regulator like a 7805 for simplicity, or a buck converter for efficiency).

A bigger problem is that the LCD you linked to requires 6 digital IO pins, which is the grand total of IO pins the Imp has. It’s really not a suitable LCD for the Imp without additional hardware complexity (I think adafruit also has an SPI board that will help you talk to the LCD with only 3 digital pins).

The Imp can send text to the planner screen or even to other places. Perhaps you could take advantage of that fact and not bother with the LCD screen.

I use the Show Input node to display various messages (string outputs) to myself.

This of course is only useful to me. I am doing project just for personal use.

Im still not very sure about the configuration of Imp with Android.
Lets say I have Imp connected, doing on/off to a relay.
Can i use the android app to connect to it, to see data about it?
Can I do that if i connect to the imp, it shows me data i decided to show, like time pass since i turned the relay on?

Im not so sure what are the capabilities of it
what is a planner screen?

About Voltages…what is the voltage needed for the imp, and what is needed for an lcd display? Will both work on 5v input?
Maybe its easier to create 5v battery other than adding more hardware which im not sure what i need
Can you show me a link for items i need, in order to convert 9v battery to whatever the imp needs

And about lcd display, Cant i use serial lcd display than? it needs one input, isnt it?