I'm unfamiliar with any Doctor Who games, except for a board game (4th Doctor) and a couple of RPG gaming guides produced for the 5th and 7th Doctors. I assume the ones you're referring to are computer games or online?
They are free online games offered by the BBC Doctor Who website, I think they were made available outside the Uk by steam or some-such. They are to date the only things that are officially stated to be Canon ( and do have continuity errors, that make them difficult to reconcile with the TV show, Ironically).
What Roddenberry did or did not consider canon is something that changed over the years. As some have said here, sometimes it depended on which company owned which rights, and which products were actually making money. I always considered TAS to be canon, which is why I cannot consider Enterprise to be canon since it violates so much of what was established in TAS (ie. Captain/Commodore April). And I applaud anyone who disavows the mess that was Star Trek V.
As with the continuity clashes with the aforementioned games, this again touches on something often missed in discussions of Canon, the difference between Canon ( which body of works are official and refered to in other stories) and Continuity ( making sure a series is internally consistent).
Continuity is not actually linked to Canon, for example both Assignment Earth and Turnabout Intruder are canon.
Continuity would be the thing that makes sure that someone disintegrated by a rogue starfleet captain in the end of the second season does not turn up in your season three finale, for example
. The fact that TAS is canon would not mean that Enterprise wasn't even if there are continuity clashes ( I have to say though I am not entirely sure how Robert April in TAS clashes with Enterprise?)
Equally separate is actual Chronology, which is the order episodes are placed and when they are dated too ( though obviously continuity plays a role in this). For example the Okuda's placing TOS in broadcast order, or placing TWOK in 2285. everyone accepts the stories as canon ( well I assume so anyway) but when and in what order they occur is up for grabs.
There is no real reason not to place TOS largely in stardate order for example, given the general trend for Stardates to increase over time, or the placing of TWOK fifteen years after space seed.
As to the Trek/Who crossover - tipping my hand to my view on Who canon... Given no canon for Doctor Who by definition it is not canon and with Star Trek, well official sources say that spin offs such as books are not canon just what we see on the screen. So its not canon. That said its a rather good story ( would love to have seen more of Kirk and the fourth Doctor interacting though) and worth any fan of both series time.