I love Doctor Who! I think a crossover would be cool, but I don’t think either franchise is a good ongoing fit for the other. The DC universe doesn’t need a character like the Doctor, and the Doctor already has a universe of endless possibilities to explore.
If a crossover did happen, I think it would be interesting if the Doctor and the heroes of DC were actually at odds with each other because the Doctor preaches total non-violence (even if they’re a little inconsistent applying it) and the superheroes always resort to punching, if they can’t get the baddies to surrender.
… Lex Luthor could be in league with the Daleks because he totally fits the bill of a Dalek coconspirator-- Got a little excited. I’ll stop now
They actually team up with people all the time, think Mavic Chen from 'Master Plan", Lyttin from ‘ressurection’, or Solomon from the Manhatten story to name a few. The Dalek human (or humanoid) coconspirators fall into two camps:
Someone who knows nothing of the Daleks and thinks they’ll genuinely share power
OR, more relevantly
Someone who knows not to trust the Daleks and thinks that THEY are the manipulating THEM.
Lex fits category 2 quite nicely, though it’ maybe a disservice to his character to have him be completely blindsided by the Daleks having a plan within a plan. I’m picturing an endless cycle of reveals like:
“But I new you were going to betray me the whole time.”
“Ah, but I knew that you knew, so planned for it.”
“Ah, but I knew that you knew that I knew, so-”
There is no way the Dr would make sense in the DCU. As much as I love them, I have to say no! However having an ongoing published by DC but not set in the dcu, i’d jump on. Also as others have said, a crossover would be good.