What should I do regarding my vow to slaughter a calf after my wife's natural pregnancy, with my intention not to eat from it and to give all of it for the sake of Allah, if I don't remember whether I intended to slaughter it alone or to make it an aqiqah? And is it permissible for me to perform the aqiqah with the ram that my wife's father will gift? And would slaughtering a calf and two rams (one of which is a gift) be considered extravagance for my newborn?

A conditional vow is disliked, but it must be fulfilled if the condition is met. If the condition is met, then you must fulfill what you vowed, which is the slaughter of the calf.

Regarding the doubt about the intention of the aqiqah, act according to what you predominantly believe, because predominant belief is treated as certainty by most scholars.

If you predominantly believe that you intended the aqiqah with the vow, then slaughtering the calf will suffice for the vowed aqiqah. If you predominantly believe the opposite, or if you do not have a predominant belief, then you must slaughter the calf for the vow.

The aqiqah is a Sunnah (prophetic tradition) and not obligatory. If you are able to perform the aqiqah for the newborn, then do so. There is no objection to performing the aqiqah with the ram that was given as a gift. The slaughter of the calf for the vow is a separate act of worship from the aqiqah.