How do you connect 4x4 Button Pad PCB + 4 Arcade Buttons to an Arduino Uno?

I can't see the picture you posted.

Not sure what you mean by buttons only responding when pressed. A button would not be very useful if it responded when not being pressed, would it?

I don't know anything about MIDI, sorry. How many Arduino outputs will be available for the buttons?

Try a 1K resistor on one of your LEDs. Is it bright enough like that? If not, you may need some transistors also.