There are a few games that do diagonal scrolling. In general it's very difficult to do well on the NES, and you will likely have to live with some amount of glitching--unless you have extra name table RAM on the cartridge, which is fairly rare. See http://bootgod.dyndns.org:7777/ for a database of the hardware inside each cartridge.
There are a few games that do diagonal scrolling. Paperboy comes to mind. It's smooth and looks good without any noticable glitching, and without using custom chips, too. Not sure how they did it.
Nope. Paperboy uses a CNROM, which is basically an MMC3. Maybe they're doing some clever edge-replacement stuff on the mirrored table, like SMB3 does with the same hardware? But you can see glitching when SMB3 does it.
(edit: my sibling post answers it. That's slick.)