I'm just going to try and get all the Sherlock plotbunnies out there before going to the major multi-chap.
Disclaimer- the characters of Sherlock are Sir Arthur Conan Doyle's, the premise is Stephen Moffat and Mark Gatiss's. Doctor Who is not mine, and belongs to the many wonderful people who write and produce for that fantastic series of fantasticness.
The Police Box On Montague Street
The landlord at Montague Street had decided that he had had enough of Sherlock Holmes the day he opened the door and found a blue police box in the flat he rented out to the world's only consulting detective. The moment Holmes returned from gallivanting about saving the world (while wearing a clergyman's uniform, apparently), the landlord gave him his notice.
"I assure you that whatever it is that happened, I likely wasn't the cause," Holmes said with a grim smile, opened the door to his flat, and shut it again. "There is a police box in my flat. Why?"
"I don't give a damn, Mr. Holmes. But I expect it out in two weeks, along with the rest of your madness," the landlord snapped, quite red in the face.
"If you need more money."
"Money is not the issue. I simply do not have the constitution for any more of your..."
The unfortunate landlord was cut off by a young man in a rumpled suit and converses walking out of the police box and placing a pair of thick-rimmed spectacles on his nose.
"Sherlock Holmes, I presume," he said with a huge smile, and grabbed Holmes's hand in order to shake it vigorously. "An honour, sir. Honestly. I have news for you. Good news."
"Do you have living quarters for me? I seem to have exhausted these ones," Holmes said with a smile that was very much trying to disguise itself as a frown.
"Uh, no. But you do find somewhere nicer, and cheaper," assured the man, and turned to the landlord. "Could you leave us alone a moment. I don't mean to offend, but you aren't important enough to hear what I'm going to say to this amazing beautiful wonderful genius of a human being."
"...One week!" the landlord yelled, and stomped down the rickety staircase like an adolescent.
"So. News. Good?"
"Absolutely. You won't be lonely for much longer Mr. Holmes. Isn't that fantastic?"
"And on what authority are you getting this information? Who exactly are you?" Sherlock said irritably. He really had no time for eccentrics outside of his work; they tended to find a refuge from their insecurities in their oddness that was not endearing, but annoying.
"I'm the Doctor!"
He said this as a matter of pride, but beyond that it was hard to read this man, as though he were processing his seemingly boundless enthusiasm and little else. Sherlock tilted his head in almost-confusion. There was something not quite human.
"What is it that you cure?"
"That's a bit beyond your jurisdiction, Mr. Holmes. You're still not important to know that. But on the 30th of January, you will meet another doctor, a medical one. His name is John Watson. Don't discount him because he knows Mike Stamford. Absolutely don't do that."
"...All right. Will you stay for tea? I don't think that your police box has destroyed the kitchenette," Sherlock said, trying to make sense of the Doctor's last exchange, and upon finding that he couldn't, stowed it away for later analysis. "How exactly did you get this thing in here?"
"Oh, she won't hurt any of your belongings. Really, you shouldn't even see her, but the cloaking device has been a bit [here the Doctor made an indescribable sound with his mouth and hands in order to illustrate what exactly the cloaking device was a bit of] and I have to find replacement pieces in another galaxy where they haven't seen a TARDIS before, so... Yes, tea would be lovely."
The Doctor liked his tea with three sugars and no cream. They talked wormholes and body farms, and then the Doctor announced that there was a woman waiting for him somewhere in Mozambique of 1847. Sherlock nodded, doing his very best not to register this piece of illogical marginalia. Before disappearing in the very visible police box that apparently travelled through time and space, the Doctor paused.
"Remember, Dr. John Watson. John H. Watson, if that makes it any easier to remember. Don't discount him. He'll save your life," the Doctor said quickly.
"And the left one is the bad bottle. Or it could have been the right one..."
The Doctor seemed lost in thought for a moment, but shrugged.
"Don't worry too hard about it. Dr. John H. Watson, remember. Bye-bye now. Good luck finding a new flat. Why don't you call up Mrs. Hudson? Nice lady, isn't she?"
The iPhone is slim and warm in his hands, and the engraving on the back seems almost raised on his fingertips.
"Afghanistan or Iraq?"