You both made a lot of mistakes, but that's how it goes in collaborative games like Hanabi. On the whole, you played all right, while your teammate was giving more nonsensical clues based on the conventions BGA Hanabi players use.
1. Players in 2-player like to give 2 clues early. You giving the 2 clue at the start would have avoided the blue 2 being misfired. Your 2 clue, once you gave it, could be considered a play clue for the other 2, since your teammate already knew the green 2 was playable.
2. Your teammate let the blue 1 slip through your hand. Twice.
3. If the newest card is clued, that means it's probably playable. Your teammate was cluing 5's early, when 5's of other colors were playable. Your misfire of red 5 was not your fault. And technically, you could have thought the other 5's were playable. Early save clues can be good, but only if the number being clued has no way of being playable. That is, a 5 clue is fine only if there are no 4's in play at all. (This advice is only for 2-player.)
4. Your teammate let the yellow 2 slip through your hand without being clued.
5. There was also some discarding of clued cards that might have had a chance of being playable, but I couldn't quite tell if I was missing out on some info (when your teammate discarded the other green card, and when you discarded the red 4).