Would anyone ever put a 3-way speaker in the dash?

If they would fit, yeah .Speakers in the size you're quoting (3.5") don't often come in the 3-way format. But if you can get one that way, sure. The more you dedicate different frequencies to different cones, the more detailed the sound reproduction becomes.

Secondly, by only putting mids and highs in the dash, wouldn't it be a bad idea not to use a crossover?

Correct. Any time you have a speaker which can't reproduce certain frequencies, it's always a good idea to use a crossover to block those frequencies.

I personally use F-mods (available from Crutchfield) to block the bass on the amp channels which go to my mid/treble speakers, and my subwoofer amp has a built-in crossover. So I am doing crossovers for all my speakers without actually owning a dedicated crossover.

Another option is to use the Empeg's ability to EQ four-ways to block bass frequencies to just your front dash speakers. If you do this, make sure to lower the Q value on those bands so you don't get notches where the bass peeks through.

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Tony Fabris