I'm not familiar with this device, but can you still get into TWRP? If so, try a different ROM to see if you can get it working again. Of course you really should have used TWRP to make a backup before you tried flashing anything: if you did that you'd be able to just restore the ROM from the backup (first rule of root club: always take a nandroid backup before modifying the system software in any way).
But what bothers me is the fact that you can't get into download mode. Because on any device I am familiar with a ROM flash should not do anything to that. So I'm wondering whether you used a ROM for the wrong model of the S4, or maybe a TWRP that was built for a different model? I'd be tempted to reflash to stock, but I think you need to get to download mode to do that (as I say, not a Samsung expert).
BTW the wipes you do before flashing a ROM (data and cache) won't remove the previous ROM. You do not need to wipe system (the ROM) or boot (kernel) in order to flash a ROM.