My question is, once I've built a project and the chip has been programmed. Can the chip be removed from the Uno board and permanently be mounted into a project, and placing a new chip onto the Uno board?
Yes.Additional info:To be able to program your new 328P chip, you either need a programmer (an other Arduino can be that programmer) or you can buy the 328P programmed with the bootloader so you can program it via USB.
No idea, the farnell website is down.An other solution is to make provisions on your board for upload via ICSP and use your elegoo as a programmer. In that case you can use 328P processors that don't have the boot loader.