However, in that discussion at that link, a point was raised in which the T7700 processor seems to have an identical pin setup (478) as the celeron 900 and the T4500, however, it is listed on ebay as a socket P, not specifically the 478 pin.
I know that there are desktop and mobile versions of 478 pin cpu's, but they are not compatible.
Would this T7700 processor work not only in this socket, but in this motherboard?
Upgrading laptop CPUs is often tricky since the BIOSes don't always support every CPU that can physically and electrically fit in the socket. There are a number of chips that can fit into Socket P and not all of them will work because there are several physically different processors (each of which likely needs its own code in the BIOS to work properly) for Socket P. Both the Celeron 900s and the Pentium T4500 are Penryn-3M-based chips with 800 MHz FSB. Any other Penryn-3M chip with an 800 MHz FSB should work as they all use the exact same silicon die. Your chipset is the GL40 which limits you to a maximum of 800 MHz FSBs. The fastest Penryn-3M with an 800 MHz FSB is the Core 2 Duo T8300, which is a 2.4 GHz part with the full 3 MB of L2 cache turned on. There is a chance that you may be able to run the closely related Penryn-6M parts with the larger 6 MB L2 die (fastest part is the T9500, 2.5 GHz) but it would just be a guess as to whether it would work.
As an interesting aside, Intel made a lot of essentially identical sockets. Desktop socket 478 for the Pentium 4, Socket 479A for the Pentium III-M, Socket 479B for the Pentium M, Socket M for the original Core Duo and some early Core 2 Duos, and then Socket P for the later Core 2 Duos all were the same general socket design. Some had a couple of pins in a corner moved around a tiny bit to prevent some of the wrong chips from fitting but you can fit at least some incorrect chips into most of the 478/479-pin sockets. I actually have most of those chips at home (Socket 478, 479B, M, and P) but haven't been brave enough to go stuffing wrong chips into my still fully functioning Socket M HTPC or Socket P laptop to see if they'll fit.
So the T7700 would be a definite "maybe" for the same reason as the Penryn-6M type chips?
It would be a "maybe" as it has the right socket and bus speed, but it is an older 65 nm Merom chip and "more different" than the Penryn-6M compared to the Penryn-3M design currently in the computer. It would be possible for the BIOS to contain code for both the Merom and Penryn chips as some Socket P laptops do take both generations of chips, especially if the computer takes the newer Penryns. My Dell E5400 is a Socket P laptop and I know that it takes all of the Socket P chips made with the possible exception of the unusual Core 2 Quad mobile chip. But the only reason I know that it is because it was available with various Merom-2M, Merom (4M), Penryn-3M, and Penryn-6M chips with 800 and 1066 MHz buses available from the factory. Your machine was only available with Penryn-3Ms from the factory if your spec sheet was correct so that is the only chip line we know would work. You could always try to get a cheap Merom or Penryn-6M chip and see if it works, I just wouldn't go spend a bunch of dough on it in case it does not work. I would also make sure that the BIOS is updated to the latest version as well.