I have the below compressor, and would like to use it for pumping my bike.
Does the below hose and gun work with it?
And what is the metal button at the end of the gun for?
If the metal button that you refer to is this one:
Best guess is that you press this when engaging the end of the gun onto and when disengaging the gun from the valve stem. When the gun is fully pressed onto the valve stem letting go of this button helps clamp the tip of the gun onto the threaded end of the valve stem.
Again this is an educated guess because I've not seen your gun first hand. You could probably look at the tip of the gun assembly and toggle the metal tab back and forth and see right away what it does.
In looking more closely at your picture it appears that the tip of the gun can have various ends slid into it and this lever is designed to hold the tip in place. Pressing the tab (button) releases the catch and allows removal of the tip.
When I refer to the tips I mean one of these from the kit:
I'm not sure what kind of fitting your compressor has. If it takes a screw-in hose, the kit you show will work with it. If it takes a standard snap-in hose, you'd need the appropriate adapter (probably available from the same place you found the nozzle). If it takes some other connector, you'll need the appropriate adapter for that.
As far as bicycle goes: It looks like the middle tip is for a Schrader-style valve, so probably, though if your tires have Presta-style valves you'll need a Schrader-to-Presta adapter. If not, bicycle/car inflation nozzles are available cheaply online or from places like Harbor Freight, and for only a few more dollars from stores which sell automotive tools. (Who will also have the standard adapters.)
Can't tell from the photo what the piece you pointed to is for. It may be an escape valve that you can trip to let a bit of air back out of the tire if you overshoot your intended pressure. It may be something completely different.
Quick observation: That small compressor isn't good for much more than inflating tires and the like. Don't expect it to drive a paint sprayer or nail gun or any other air-powered tool.