Hi Boshk,
Remote controls for the amplifier often have facilities to control a matching CD player, I like keeping the wand count down (I wouldn't make it in Hogwarts, I know). The next point is more conjecture than knowledge but I would suppose manufacturers, knowing the designs of their amps inside out, would design their players to integrate well with other components. I could be wrong though.
As for your second question, I think that depends in part to the DAC's employed in both the amp and the CD player. Unfortunately, no, they are not always the same. For example, I'm looking at the Onkyo reference preamp and reference CD player; the preamp uses the 1795 and the CD player uses the 1792 if I recall correctly. All other things being equal, the 1792 is 'supposed' to be the better of the two so I'd lean towards an analogue interconnect. That being said though, I would try out each option and decide which one I preferred on a case-by-case basis.
On a more commercial note, the CD player has been around for quite some time. One of the things I have noticed quite a few firms doing recently to maintain their portfolios is to use the CD player case to serve as a digital input source array. In the absence of all the extra components involved in preamps and integrated amps, it allows for higher-end DAC's to be included with the player than the amp. But of course, I generalise - you would be best off trying the combinations you are considering both via digital and analogue connections.
Still, aesthetics and a single remote are powerful drivers in my house!
Cheers
LDTM