Compared to single player mode, do enemies in co-op mode (both Campaign and Whored mode) become tougher or deal more damage? By how much? What about their numbers? I'm also curious about what changes in missions, including the DLC missions.
I don't think so.
I believe that the amount of enemies also remains the same.
A few ways that I know that the difficulty increases:
While it may not be exactly what you are asking, in my experience co-op campaign is significantly easier than single player (although neither was particularly challenging at higher levels, due to unlocked buffs). I never noticed an increase in enemy damage or quantity for missions, timed gang battles, or just free for all fights in the street.
Instead, when you've got a buddy you always have the power to split up enemies, pack twice the firepower, and the ability to bring a superhuman from later in the game to fights earlier in the plotline where your character is much weaker. Recall that by end game your character is basically unkillable, so having such a character for a pal can really never be called "harder" in my opinion. Even if your friend has not yet become literally bulletproof they may still have power vehicles and weapons which you can make use of - such as the ability to call the helicopter or VTOL homie.
Also, there are a number of "Co-op" style missions with the helicopter or the flame retardant suits, which are actually very challenging when playing solo - simply because of targeting difficulty while driving a vehicle. Having a co-op companion for these missions is not only helpful, but for the highest levels may be borderline required - not really because of enemy hitpoints or quantity, just because of timed missions and low tolerance for targeting errors.
The odd case of "Whored Mode"
In a riff on Gears of War, SR3 includes a separate horde mode with totally different rules. For deep details, see the wiki. A few key aspects are:
I do not recall any difficulty scaling associated with Co-op "Whored Mode", but if there is any it's not really noticeable. As a side note, don't waste too much time in this mode - the campaign is way more fun.