timremy, maybe we should go back to square 1 (well, 2 or 3) here. Not trying to talk down to you but we're all making a lot of assumptions about what's happening.
Since you can boot into cloud mode can I assume you have Tiny Core Linux on a CD? And that you have your USB device plugged in when you boot? And that booting off CD works OK, with no error messages?
Can you see the point in the boot process where you can enter bootcodes? This is the instruction where it says something like 'Press enter, or F2/F3 to enter bootcodes.' At this point you should press F2 or F3.
Type [no quotes] 'tinycore waitusb=10 tce=sda1' and press Enter. This tells Tiny Core to give your USB device time to mount and that you want your tce directory (which you need to save extensions) on your USB device. I'm specifying waitusb=10 in case you have an especially slow device (waitusb=5 is more common but you tried that).
Once the boot process completes you should be at the Tiny Core desktop and see a minimal set of icons at the bottom of the window. The one on the left will give you a terminal window, the middle various utilities, and the right (gear looking thing) is the app browser which lets you load various extensions (web browsers, media players, etc) into the tce directory.
To verify a tce directory exists, start a terminal window and type [no quotes] 'ls /mnt. Among other directories you should see sda1 [or at least sd with a letter and number]. If you don't, your USB device is not mounted. If you do, type 'ls /mnt/sda1/ [or whatever designation your system gives your USB device]. You should see a blue 'tce' directory entry at this point. If you don't, type 'mkdir /mnt/sda1/tce' [again, use the name your system gives your USB device] which should create a tce directory.
Reboot here. When booting, again use the bootcode waitusb=10 so your USB device is mounted. Tiny Core should scan your storage and find the tce directory.
At this point you should be able to download apps and have them automatically stored in your tce directory. Note that this is not the same as installing Tiny Core Linux on the USB device so you can boot from there. That's a separate process which others here are better qualified to help you with. [My PC won't boot from USB and has a clunky hard disk, so I'm still booting off CD and only using USB for storage].
Hope some of this helps!