The likeliest thing one has to suggest would be when you replaced the 1st hard drive with the new one, that you did not make the appropriate settings on the new hard drive vis-a-vis Master/Slave/Single. This usually causes what you are seeing in regard to the other hard drive.
However, it should not in any way at all affect your DVD drive. THAT is a bizarre aspect. The cables connect pretty closely on the motherboard and one might suspect you had fumbly fingers or along the way pulled both cables and, along with not setting the new drive for proper Master/Slave/Single, might have not reconnected the DVD drive cable properly.
But then nothing you do other than seating that cable should make it show up and function! Not anything with Device Manager, a Ouija board, or prayer. Nothing but properly seating the cable.
But the cable proximities give me a thought. The caveat is I've never seen this in practice and am unwilling to bet (more than $5) on it.
So, I'll go out on a limb and guess something like the following:
When replacing the hard drive with the new one, I think it possible you, somehow, and it would probably take a lot of "somehow" to do this, connected the new drive to the DVD drive's cable. Each cable takes two devices, and then they have to be set, properly, for Master and Slave. (The usual motherboard has two controllers, with their cable connectors very much next to each other and each can control two drives.) So, even if you configured the new drive to its likely proper role of Master, you have problems: the DVD drive is also configured as Master. In addition, the other controller then has a single drive now, but it is still factory configured as Slave, not Single.
So in the start-up process, the far-less-capable-than-Windows BIOS routines fail to find the 2nd drive, mis-configured as Slave, altogether and when asking the other controller for what's there, it gives the answer of the 80GB hard drive (perhaps it asks the cable and the hard drive responds the fastest so it gives that as an answer even though the DVD drive chimes in claiming to be Master as well; perhaps it gets the DVD drive's reply but that is overwritten by the hard drive's reply and given as answer; perhaps it considers both and chooses a hard drive over a DVD drive — it doesn't matter, for whatever reason, it tells the BIOS only about the hard drive).
Then, since the BIOS knows only about one hard drive, it seeks to boot from it, which is actually the desired outcome. This masks the problem since all seems to be working fine. Windows boots up though and takes the BIOS's "word" as sufficient and fails to list either missing resource.
However, when you tell Device Manager to refresh its listing, it seeks all new information using much more complex/complete programming than a BIOS ever could, finds all devices reporting in, and lists them, as well as assigns drive letters. Which, assuming you didn't do anything relevant first, would be exactly the drive letters you would have wanted assigned so all now seems right with the set-up. Except that it needs that step!
A quick way to test for this might be to boot up the computer from cold iron and, before even opening Device Manager, round up and pop in one or more thumbdrives or camera/MP3 cards. Check My Computer for drive assignments. Likely, even if it detects the errant drives while doing this, it would assign them letters after it assigns the first thumbdrive letter. So the hard drive might be E: instead of D:.
If so, or if it still does not list the errant drives until you refresh Device Manager and then gives them the different than expected drive letters, I'd raise my bet substantially.
At that point, or without doing it but rather right away, you could open the computer and check the cables. The two hard drives should be on the same cable. It is an amateur's mistake to think they should be on different cables, but you seem to fall into that category. Both belong on the same cable! Ensure the DVD drive is on one cable, and the two hard drives are on the other. The 80GB hard drive must be configured as Master since the "2nd" of those 250GB drives would have been Slave (or you would have been calling IT the 1st drive). It's usually done with miserably small little beasties you place just so. However it is done though, make it match the manufacturer's information. Be sure the two hard drives are on the same cable and the DVD drive is on the other and that both cables are nicely seated in all the connectors.
Then fire it back up and, perhaps, everything will now play nice-nice.
If not, then I'm at a loss. Usually problems that come after a single event stem from that single event and often from one, sole mistake during the event. I do not see how the 1st drive's failure could have laid the seeds of this failure in the computer so I believe it happened in the recovery phase, especially if an inexperienced person did the actual work involved. The new drive might have been configured properly, but have caused the effect with the sole mistake of being put on the DVD drive's cable. That would bring us down to one, single, sole cause for the whole situation.
Good luck though!