I think most of you seriously underestimate Littlefinger. I seriously doubt he will accept so closely people he doesn't know or he doesnt't trust. The idea that The hound is ser Byron is absurde. In the books I think only Melisandre and Mance have such transformation and Howland Reed even a crannogman is not a magician. To transform The Hound in Tyrek Lannister... It's not the kind of fantasy of GRRM. Too "fantasy" I suppose.
I think Littlefinger knows who ser Shadrich, Morgath and ser Byron are, because he's "hired"them. If ser Byron is Tyrek Lannister I suppose Littlefinger has "a finger" in his disappearing from Kings Landing. And maybe Morgath is some knight that helped Tyrek to escape from Kings Landing and that baby-wife he was mocked for. And having Tyrek Lannister close is useful for Littlefinger in his "game of thrones". Sansa is just a child her head is full of songs and fairytales and she usually sees only the obvious and as a child she is naive. So I think it is possible for her not to recognise Tyrek with no Lannister signs and after two years. I think the three hedge-knights are not there to save Sansa from Littlefinger but to serve Littlefinger.
About ser Shadrich I also have doubts thet is possible to be Howland Reed. When he meets Brienne he says that he was at Blackwater battle on the wrong side and lost everything. He's not a tourney knight but he's a skillful warrior in battle. He's a knight. He has the behavior of a nobleman. When I see the list ot nobleman knights from tne Battle of Blackwater on the side ot Stannis i see the name of Bryce Caron. Bryce Caron the orange. In the book there is no description of hif physical atributes. But he has orange hair, so is Shadrich. He took part of few tourneys but win none. After Battle of Blackwater he was "killed"by the unknown Philipp Foote and Foote was awarded wit all Caron lands. So Bryce Caron lost everything, just as Shadrich said he lost everything. Bryce Caron lands are in Stormland marches near the border with Dorne. It is similar as Shady Glenn. You connect mouse with bat and Harrenhall and House Whent. I think the mouse is metafore of a man who lost everything important and significant and who become no one. So for me is possible Bryce Caron is alive and in service to Littlefinger. Littlefinger has the power to set free and use a captive. I think it is possible Bryce Caron is not killed but captured. And this Philippe Foote is just a strawman a puppet of someone other's interests. Maybe Littlefinger's. The waekness in this theory of mine is that Bryce Caron was one of Rennly's seven with Brienne and I'm not sure how she didn't recognised him. But may be there is possibility that she doesn't knew him close.