Both processors are pentium 4. The motherboard has no identifying marks on it.
Possibly/probably. Intel made several 2.53 GHz and 2.80 GHz P4s, but fortunately most of the different generations of parts (that wouldn't be compatible) have different sockets and won't fit in a board that won't support it. As a general statement, if it fits, it will likely work, although it might run slower than 2.80 GHz (e.g. putting a 800 MHz FSB 2.8C chip in a board that only supports 400 and 533 MHz FSB chips.) The only ones that will likely cause a lot of trouble are the 2.80 GHz 478 Prescotts.
But as we've said above, if you can take and post pictures of the tops of the CPUs in question, we could probably tell you for sure.