The old HDD issue
should be simple, or at least doable. Unless you've had it used in some sort of encrypted/secured format which (SFAIK) isn't a standard installed configuration except for maybe some laptops.
Exactly what to do (hardware-wise) depends upon if SATA or PATA(/IDE). If you've got space in the case. If the Mobo has enough native cables (though there are ways to get around that).
If it really isn't possible to do that, caddies/drive-docks exist that make the old internal-HDD
directly into an external-HDD (USB or perhaps eSATA, generally).
No need to reformat the drive, although I'd suggest that (before you use it as a D:/E:/F:/whatever:-driive) you get some sort of Drive Image of it (live Linux-tools bootable DVD packages abound, from which you can create one or other) and store a copy or two on DVD(s)/wherever so that you can hopefully restore it if it ever becomes necessary, then you can aim to go through all the OS-level gunk and old My Documents-type stuff and either shove it into a bitbucket or make use of it as you see fit.
I mean, I tend to just
let copies sit around forever. I'm also not very good at cleanly migrating between my own old/new machines.
User data acess is a trickier thing to guarantee.
Browser passwords I would (depends upon your browser how you can do this) export or review the details of and make a note of, because you might not be able to import straight from the system-level files that it has taken pains to have securely encoded such data in exactly so that nobody
can recover such details. Work out how you'll have to get it into an importable/re-enterable format before you actually rip the old machine apart, because some things might not be trivial from just the drive contents being loaded into a new systems stack of storage devices. I mean, can a "remember my password cookie" be necessarily transferable? What if you've got a Password Manager of some kind that is constructed to not to have its details stolen from by any idiot with a thumbdrive? Think ahead.
Maybe savegames can be copied directly from the old drive's installed directories into the ones the new installation on the new machine has made. But I wouldn't guarantee that's possible. (And, though it works perfecly Ok like this for DF, in general trying to run the old-install from the old-drive location is probably not going to work as well as you'd want, if at all, for various reasons.)
Sight-unseen, of your particular situation, I might even suggest that it could just be simpler just to keep the old machine 'running' (turning it on only as you need to, to refer to it) rather than rushing into extracting its old drive. If it's still working where it is then you can get used to your new system and then dive into the old one as and when you need to to copy (by eye or by USB drive) whatever you want transfered into your newer computer. After a while (and having made the aforementioned drive-image!) you can finally decide that there's nothing at all new on it and
then rip it out, completely repartition/reformat it and use it as a handy further drive.