If I search for a counterspell and an Elspeth, sun's champion with the jace ultimate can I counter one of their spells on their turn and then resolve the elspeth along with it at instant speed on their turn? Thank you.
No.
Since you can only activate planeswalker loyalty abilities at sorcery speed, there is no way for your opponent to have a spell on the stack that you can counter with a counterspell you grab with Jace's ultimate
No. Jace's ability directs you to cast the spells you search for immediately, as an effect of the ability resolving. In most cases, this means that if you take a counterspell from someone's library, you will only be able to cast it targeting one of the other spells you cast off of the ability, because planeswalker abilities can only be activated at sorcery speed and as such there would never be anything below them on the stack. (Pretty much the only way I can think of to have this not be the case requires a setup involving Rings of Brighthearth, Strionic Resonator, and precise timing that would be unlikely to play out in a real game.)
If the ability gave you permission to cast the card later, it would state a duration to indicate that it's a continuous effect (i.e. "Until end of turn" or "As long as those cards are exiled").
If I search for a counterspell and an Elspeth, sun's champion with the jace ultimate can I counter one of their spells on their turn and then resolve the elspeth along with it at instant speed on their turn? Thank you.
Since you can only activate planeswalker loyalty abilities at sorcery speed, there is no way for your opponent to have a spell on the stack that you can counter with a counterspell you grab with Jace's ultimate
If the ability gave you permission to cast the card later, it would state a duration to indicate that it's a continuous effect (i.e. "Until end of turn" or "As long as those cards are exiled").