Just a rought translation (fell free to correct anything...I'm nor japanese nor a native english speaker ^^")
Yuzuru/Yuna conversation :
Yuzuru : Do you remember (me/the old time) ?
Yuna : Yes
Yuzuru : So you remember...that's great. Keep at it.
Yuzuru : [to cameraman] : I totally think she is more skilled than me with L + To jump (laught) Try your best.
Yuna : Thank you
Yuna seems so happy to see Yuzuru. And Yuzuru, really, how could she forget you ?
For Tsuzuki sensei, well, basically he said that Yuzuru and Yuna have common point : Yuzuru hates losing (and so is Yuna), they are "complete type" of people. [not sure how to translate the "完全主義者" in english. Someone ? ].
And as Yuzuru, Yuna aims for things and keep going for her goal.
I think "完全主義者" (or "完璧主義者") can be translated to "perfectionist".
It is someone who sets a very high level goal or standard for himself to achieve and does't allow himself to make any mistakes. (sounds like someone we know,right?)