correct - door speakers are only 5 1/4" mounted in a "tuned port" to make them act (using the term loosely
) like a "woofer".
The factory amp indeed has a crossover built in which only sends lows to the door speakers (which incidentally is a combined signal from the front and rear channels of the head unit). I don't know the cut-off point of the factory amp, but it doesn't much matter w/ the factory speakers. It's nearly impossible to get good bass from 5 1/4" speakers when driving them w/ 40W max.
If you install full range speakers in the door and want full range from, you
will have to replace the amp. Otherwise, just run the factory leads into the crossover (optional) and/or only connect to the woofer leads.
Having BTDT, don't waste your time trying to replace the factory speakers w/ other 5-1/4 to improve the low end unless you also upgrade the amp. I tried that route w/ 3 sets.... 2 sounded worse than stock, 1 was as good as (which ain't sayin' much). Instead, make a custom wood baffle (to bolt in the OEM holes) and mount either 6-1/2, 6-3/4 or 6x9 super efficient (2ohm, at least 92dB sensitivity, and below 45kHz) speakers.
Here's a template for the baffle
I used 6.75" but if I were to do it again, I'd probably go to 6x9 (yes they will still fit - just need to turn/adjust them a bit on the baffle) to get even lower freq responses.