"Access is Denied" Error Message

I'm using the latest version: 0.868

Make sure you do not download your files onto a system folder or a removable device.
eMTee suggests this article as an answer to your question:
FAQ #92: “I can't find my downloaded files or settings. Where are they?”.

All files are directed to a folder I created outside of the system and program folders.

Including the Unfinished files folder?

Yes, both finished and unfinished go to the same location.

This error happens specifically when a download starts and the program tries and fails to create/open the temporary file where the download stored until it finishes.
Check the path and try to create a file with the same name you trying to download there manually. Also make sure the user account that runs DC++ has proper rights to write to the folder and that the full path, inlcuding the file name you trying download does not exceed 248 characters.
Hope this information helps.

This is absolutely bizarre. I've tried changing it over to downloading to the desktop and a folder within the original folder I created, but it wants to keep downloading on the path I originally said. Guess I'll have to try reinstalling and telling it to go to the desktop first since it doesn't want me to change it later.

Also, whenever I try to download a filelist, it automatically wants to send it to the program folder. I don't see anywhere to change that.

It's a new computer and there's only one user account, so I don't know why it wouldn't have permission to write to a folder I created and already filled with several other files. I've never had this much trouble with DC before. Why would it work perfectly the first time, but then just stop working when opened the second time?

Probably also because the UAC redirection service, which can also affect the settings files. It always make programs read the old file in the system folder while it only allows writing to the redirection path protecting the system area.

As you said you have used <only> the installer to install DC++ to a clean system, I'm not sure how this could happen in the first place. The installer (not the portable version) uses defaults which follows the Windows norm, it puts every settings file and sets the download targets to the user profile, as every modern program expected to do.
Read more: https://dcplusplus.sourceforge.io/webhelp/faq_install.html

Reinstall is a good idea, but only if you check the folders, including the path the UAC Redirector service uses (shown in e.g. here https://www.sevenforums.com/news/19426-user-account-control-data-redirection.html) manually, after running the DC++ uninstaller and delete every possible remnant the uninstaller may not aware of.

More help in DC++ settings files guide - https://dcplusplus.sourceforge.io/webhelp/faq_settings_files.html

Well, it looks like everything is working perfectly now after I selected a different option during the install (something involving the user profile) at the part where it also asks about the 32-bit version. Not sure what it did, but it seems to have done it.

Thanks for your assistance eMTEE!