Assuming they are on opposite teams, if Fiora uses her ultimate on a Shen that is channeling his ultimate does the Fiora end up with the Shen at his new location?
Yes, but only if Fiora has vision on Shen's ultimate location. The latest Rift Myths just answered this question for us: If Fiora uses her ultimate on Twisted Fate while he is channeling his ultimate will she teleport with him?
It turns out it is based on if you have visibility of where he is teleporting. The first test shows that Fiora will not move with TF if she does not have vision on his final location. The second test she does move because she has vision on the final location.
This also lines up with what Micheal Ayers tested with Nunu.
With regard to the other answers tests and new found description
The League of Legends wikia has other description for Fiora's ultimate that says:
This description, if correct, just say it all.
Case 1: You target a hero that is not Shen,
then you will hit 5 targets in range of this hero that may include Shen. If Shen is in range, he will get instantly hit. If by any synchronisation odds Shen did port just between the target selection and the hit, we can assume he will still be hit at his new position, but Fiora will get back to the original location as soon as she selected the new target which will be in range of the first initial target.
Case 2: You target Shen ultiing.
You get 1 hit on Shen, then 3 random hits on targets around Shen. In the same manner if there is no synchronisation odds you will hit only targets in the battlefield where Shen is. If you have a synchronisation shift you may hit one target on the first battlefield when Shen has already moved. All other targets will be in the second battlefield.
It is impossible to answer this question. All depends on the implementation of the algorithms by Riot Games.
Nevertheless we can state that Fiora should not select a target as valid if it is more than "Range: 400" away, which is her ultimate range. The other rule in Fiora's utl says that the first and last target is the same. If Shen gets ult first, that rule may override the range limitation.
Good to be know, as Janna I once casted my bird on a recalling champion, he ended up heading toward the fountain despite the range.
With regard to these consideration, I will assess that Fiora could port with Shen in the following conditions:
I think that Shen being regarded as a valid position just before he finishes TP is not a valid case because that would mean that Fiora's algorithm is looking for Shen position gets the old position, acknowledges the position, then retrieve the position AGAIN and gets the new one. That could happen only if the algorithm is implemented as this: select a valid Champion then ask for position. In this case, the synchronisation could end up with a target being "followed" while porting.
Anyway, only insight of the source code or testing could reveal the behaviour.