Is Karate Champ, or other dual 4-waygames, impossible to play with 8-way joysticks?
They will not be IMPOSSIBLE, but they will not play WELL.
Are there many games that require digital joysticks?
Standard joysticks ARE digital joysticks.
Are games that require rotary joysticks, like Ikari Warriers, impossible to play with an 8-way joystick and a spinner?
You will not like trying to play these games with a spinner.
How about these 49-way joysticks and "perfect 360s" that I've been reading about? What's the benifit of these?
No_One already covered the 49-ways.. they'd make a good replacement for your 8-ways, since they do 8-way, 4-way, 2-way, etc. You will need GroovyGameGear's new 49-way interface, one per stick to get this functionality.
Perfect 360s are super-fancy digital sticks. They will not enable any games that you cannot play with regular sticks. You may prefer them for other reasons: They're quiet, reliable, and have a good feel during gameplay. (heavy spring recommended)
You might also consider the T-Stick+, or the Prodigy, as replacements for your 8-ways. They switch between 8-way and 4-way operation, without opening up the control panel and messing about. They don't cover 49-way, or low resolution analog like the 49-ways, but they don't require a special interface, either.
Can you use an analog or 8-way joystick for Atari games that used a "Hall Effect" joystick?
No_One already covered why you don't want to use an 8-way on these. An analog will work well on games that used an analog, hall, or 49-way joy. A 49-way will work well on games that used 49-ways, pretty well on games that used a hall, pretty well on some that used an analog, and lousy on other analog games.
Games I know will be next to impossible to play include:
720