Can't download custom APK files on Cyanogen Mod 7
Summing up the solution from above comments on the question itself:
Finding the culprit
First step is to get to the logging information on your device, to figure out some more details on what's going on. Multiple choices to do so:

get a logcat app from the Playstore. There are many, and as long as you run Android 4.0 or below, none of them have issues. Starting with Android 4.1, apps can only see their own log entries (so a logcat app only sees what it writes itself, which makes it quite useless) -- unless your device is rooted (your CM7 is), and you convert the logcat app into a system app (which can be done e.g. using Titanium Backup ★ root). To find a suiting logcat app, simply use this playstore search -- AFAIK none of them comes pre-installed with CM.
get it done from your PC. See Is there a minimal installation of ADB? for how to get ADB installed on your Linux/Mac/Windows computer, and how to set up its configuration so it recognizes your Android device. Then either use adb logcat from the command-line, or a graphical interface (e.g. QtADB, as described in the linked question)

Solving the issue
As you followed above device, you were able to pick the corresponding error message from the logs: the app was looking for a folder named /mnt/sdcard/download/, which did not exist -- instead there already existed /mnt/sdcard/Download/ with a capital D. The problem behind this little thing is: While the underlying file-system (FAT) supports mixed-case file/directory names, it does not differentiate correctly: cd download doesn't work due to different spelling, but mkdir download conflicts, though. So the solution, in your case, was to rename the existing Download directory to download.
While I still hold this as a bug in the apps used (they should be able to work around the issue, at least by asking the user where to download to), this simple trick at least enables you to download the .apk files now :)