1. Explaining Why Ten Stopped Running
It's a reasonable guess that for the Tenth Doctor, these events take place after 'The Waters of Mars', since he is sans-companion and mentions meeting Queen Elizabeth I during this period. Having had his death prophecised some time earlier, he's running away from Ood Sigma's summons to the events that he believes will result in his demise. Something I never understood about 'The End of Time' was why the Tenth Doctor who could have kept running forever suddenly decided, without any apparent prompting, to face the music on the Ood Sphere. Like the War Doctor, the Tenth Doctor would forget the events of the episode. But since he remembered meeting (among other things) Elizabeth, we know that some of the experience stayed with him - not enough to pollute the timeline, but clear enough that he doesn't seem to notice anything amiss about the encounter. Running from what he believes to be his true death, it must have been reassuring to encounter his next incarnation who must have survived whatever Ood Sigma has for him. Though the memory would fade, maybe that reassurance stayed with him subconsciously and gave him the courage to set course for the Ood Sphere. What if Eleven, knowing what his predecessor was about to face, bequeathed a wicker hat (not unlike a stetson) to his younger self the one that we see Ten wearing when he arrives at the Ood Sphere as a symbol that his future is in safe hands, which he can take with him to face his uncertain fate. This would have been a fantastic second send-off to David Tennant's Doctor and a fanservice moment that could only have been more perfect if Ten had been wearing an actual stetson in his final episode.