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For the first question, since it's outside the stakes (or spray-painted line), I believe it is treated as casual water and you get a free drop no nearer the hole. I think your partner is right on the second one. Since he didn't find it, it's a lost ball.

EDIT: Here's language directly from the Rules of Golf:

I guess I was wrong on the second one.....from Rule 25-1:

c. Ball in Abnormal Ground Condition Not Found
It is a question of fact whether a ball that has not been found
after having been struck toward an abnormal ground condition
is in such a condition. In order to apply this Rule, it must
be known or virtually certain that the ball is in the abnormal
ground condition. In the absence of such knowledge or certainty,
the player must proceed under Rule 27-1.
If it is known or virtually certain that a ball that has not been
found is in an abnormal ground condition, the player may take
relief under this Rule. If he elects to do so, the spot where the
ball last crossed the outermost limits of the abnormal ground
condition must be determined and, for the purpose of applying
this Rule, the ball is deemed to lie at this spot and the player
must proceed as follows:
(i) Through the Green: If the ball last crossed the outermost
limits of the abnormal ground condition at a spot through the
green, the player may substitute another ball, without penalty,
and take relief as prescribed in Rule 25-1b(i).
(ii) In a Bunker: If the ball last crossed the outermost limits
of the abnormal ground condition at a spot in a bunker, the
player may substitute another ball, without penalty, and
take relief as prescribed in Rule 25-1b(ii).