For starters, write down all mounting commands you used in DOSBox.

Oops. I'm bad at this ^^;

mount e e:\ -t cdrom -usecd 1 -ioctl

I'm also using Windows XP and DOSBox version 0.65, if that helps

A little more specific...

The problem occuring is that the installation process won't even start. It'll say that a certain directory doesn't exist or that access to it is denied. This has happened 47 times... today.

My computer itself is new, having just gotten it 2 days ago, but the specs are somewhat low. It uses an Intell Celeron D Processor 352, has 100 gigs of hard drive space, and has 256 MB DDR SDRAM

It seems you mounted the CD-ROM drive but you forgot to mount the hard disk. Create a folder named, for instance, oldgames using Windows explorer and then type in inside DOSBox:

mount c c:\oldgames

mount e e:\ -t cdrom -usecd 1 -ioctl

*** MOVING TOPIC TO DOSBox GAMES/APPs. by the way... ***

Thank you. I'll give it a try

Okay. That allowed the download to start, but now it's saying the game needs 0 bytes of disk space, and that only 0 is available, then stops the download (it stays on the download screen, however)

Having tried adding on "-freesize" to the end of my E: and C: mounts, it's still doing the same thing as before and saying there is no space available

You have to add a numeric value (in Megabytes) after the freesize.

-freesize 450

I did. I kind of guessed you would've assumed that ^^;

Well. I am beginning to feel I am out of ideas. You could try installing the game from windows (natively). After that (if succesful), try to reconfigure sound when using DOSBox. Anyway, the NTVDM supports basic SB 2.0 sound, if I recall correctly.

Did you try creating the directory for the game yourself and then installing to it through Dosbox? Some games can't create a directory sometimes...

Yeah, I tried creating a directory manually, but that didn't help

Indeed. I get the same problem here, even in CVS. My copy of the game is from "Player's Choice" budget release, probably identical to the original release. The host system is completely different so it must be something in DOSBox itself.

EDIT: Apparently, there are a few problems with this game. I've installed it in VirtualPC (the game was not particularly happy there either, mostly because of amount of available RAM, but I did not dig in that direction) and copied the installation to DOSBox. At this point there was some odd unrelated problem that DOSBox refused to list files in that folder (failing in function stat trying to convert creation time to system time??? -- I will have to follow up on that). Corrected the issue and got two more. The game installer does not detect Soundblaster, and the game immediately exits with cryptic message. Hmm, too many problems for one game. Let's see if I can figure those out. Oddly, there is another game based on similar engine (Rex Nebular) and that one runs without a glitch.

I couldn't get a native installation to work, either. I even tried installing it with a compatability program running for it. I'm out of ideas ^^;

No good new here 🙁 I've digged through other forums, there were a few posts about Dragonsphere before. The reports were spotty -- some people reported the game working but it looks like they were referring to floppy version (in all likelyhood, downloads from grey sites would be of either floppy version or zipped install). So, I've debugged a little deeper and that's were I've got some bad news. Right before showing that dialog the game calls int 21h function 52h (Get List of Lists). In less technical terms, tries to get some internal information that no game really should be accessing. If I understand correctly, List of Lists is not a complete implementation yet. There are some things there that differ from real DOS. Well, this is not good. I had Dragonsphere on my list of games to play (never had chance to finish it before) and I was sure that it is working in DOSBox.

Guess they are accessing the DPB to query the freesize, but those tables
are highly drive-geometry dependent which is impossible to implement
for the directory mounting of dosbox.
If you find that check, you might want to hack it out of the executable.

Dragonsphere CDROM w/full speech under DosBox 0.65
Dragonsphere CD-ROM w/ full speech under 0.65

Thanks, that's very useful. Curiously enough, it looks like your method is not just a workaround but rather an attempt to restore the game install to the original state. Let me explain what I mean. After messing with that "Player's Choice" edition I've got an impression that whoever prepared it did not know the stuff very well. The entire graphical setup looks like a bad hack. Directory configuration looks strange, to put it mildly. The last item -- why is one of the folders called "ROMZON"? It almost sounds like they've DLed it from some pirate ftp... I just hope not.
So, I went into trouble of acquiring a copy of different edition. Got it yesterday in mail, the original UK edition shaped like a small book. Quite a nice package, very becoming for an adventure game. Guess what: the setup is completely different. As a matter of fact, it is very similar if not identical to floppy version setup. Surely enough, it worked without a single glitch, and the game works perfectly afterwards.
Time permitting, I will still try to figure out the problem with the "Player's Choice" installer -- that edition is quite common and likely shares the issue with other "Player's Choice" releases (it's marked as #7, so there are at least seven of them).

I must admit, I don't have an original of this game (where did you get yours from, do they ship internationally?), but when I tried installing I didn't get any errors.
I figured it was because it was tampered with... but I guess it's just a copy from the original...