13 July 1977: birth of Joe Lidster, writer of A Day in the Death (Torchwood, 2008), The Mark of the Berserker (Sarah Jane Adventures, 2008), The Mad Woman in the Attic (Sarah Jane Adventures, 2009) and The Nightmare Man (Sarah Jane Adventures, 2010) as well as lots of other spinoff literature and audios. I find him one of the best New Who writers, and am a bit surprised that he has not yet written for the main show. Recently we have been bantering on Twitter about the 1970s series Secret Army, whose first episode was shown eight weeks after he was born. (I'm a bit older.)
13 July 2002: death of Carey Blyton, composer of incidental music for Doctor Who and the Silurians (Third Doctor, 1970), Death to the Daleks (Third Doctor, 1974) and Revenge of the Cybermen (Fourth Doctor, 1975). Nephew of the more famous Enid.
ii) webcast/performance anniversaries
13 July 2001: release of the pilot webcast episode of Death Comes To Time, the weirdly canon-violating and very long Seventh Doctor story, basically an audio with animation.
13 July 2013: First performance of that year's 2013 Doctor Who at the Proms.
iii) dates specified in-universe
13 July 1643: the battle of Little Hodcombe, and surrounding time-travel events involving the Fifth Doctor, as shown in The Awakening (Fifth Doctor, 1984).
13 July 1964: the events of the earlier phase of Whatever Happened To Sarah Jane? (Sarah Jane Adventures, 2008) - was it Andrea or Sarah who drowned? The best of the first series of SJA for my money.
It is an interesting historical footnote that in July 1964, as the fictional Andrea and Sarah are reading about the Beatles playing Brighton, the real-life Jane Asher, who plays the grownup Andrea of 2008 (on the right in the picture above), was Paul McCartney's real-life girlfriend. For comparison, pictures of her then and of Francesca Miller playing the young Andrea:
13 July 1977: birth of Nina Rogers, protagonist of the short story "Consequences" in the (excellent) Torchwood anthology of the same name. Not terribly surprising that she shares a birthday with the story's author, Joe Lidster.
13 July 2029: The Sixth Doctor and his companion Constance (played by Miranda Raison) arrive in Arizona for the events of 2015 Big Finish audio Shield of the Jötunn (by Ian Edgington, directed by Louise "Leela" Jameson).