You can remove an ATMEGA from an Arduino and use it in your own board, however when used on a typical Arduino board the chip will be fused for an external crystal as a clock source, rather than the (simpler but less accurate) internal clock.
You cannot change the clock source over ISP unless the clock source the chip is currently expecting is operational, so if you want to change the clock selection you will want to do that before removing the chip from the Arduino.
However, you can also provide a clock crystal and its two capacitors (or a substitute resonator which often has this built in) on your external board.
Typically chips (even those pre-flashed with a bootloader as a convenience) are cheaper than boards, so it's unclear why you would want to move the same chip off the Arduino as opposed to using another. You will want a USB-logical level serial adapter anyway, and once you have that you can do development on the bare chip on your board, even if you do initial testing on the Arduino. In a pinch if you don't have the USB converter you can either pull the chip out of an Arduino board and run wires from its D1 and D0, or hold the on-board chip in reset to allow you to borrow those serial lines without interference.