What does "in" do here? How come "in" is necessary here?

"feet" is used as a unit for length in English? Like for how tall a person is, and for how deep a cave is?

That's right! It's part of the "imperial system" of measurement. So 1 meter in the metric system is 3.4 feet in the imperial system. This system is mostly used by 3 countries: the USA, Liberia, and Myanmar. But It's not uncommon to hear it being used to refer to height by other people in different countries. Ex: I'm five foot seven. Ex: Most wells are 100 to 800 feet deep.

What does "in" do here? How come "in" is necessary here?

Good question. No, "in" is not necessary to use here and sounds a little bit awkward. He most likely meant to say "as an apology" instead because this is more usual. However, he could have been using "in" to describe the purpose or way by which he gives him something, which is in the form of an apology. However this would still sound weird because he would have omitted "the form" in this case. So it is best not to use "in" in this context and use "as" instead here.

What does "getting" mean here?

"Getting" here means "receiving". She is saying that she thought she would receive a female doctor. Here are some examples using "getting" to mean "receiving". Ex: I am getting a package today. Ex: She is getting her nails done.

Since she already did play the role, wouldn't to use the past tense "played" be more right to say?

Yes, if this conversation were taking place right now, then it would be correct to say "played" instead of play. However, in the case of this video interview, the series was still ongoing, so her role as Helena had not actually ended yet. If someone asks her the same question now, they would say "played" (since the show has ended and she no longer plays this role.) Ex: I played the role of Santa Claus for Christmas last year. Ex: Chris Hemsworth plays the role of Thor in the Marvel Universe.

What is the meaning of "go" in this sentence?

"Go" in this case means to try something. So "one go" here refers to "one round", "one try" or "one turn". A common way to use "go" in this context is in the expression "give it a go" which means to try something or have a turn at doing something. Ex: I tried to give the bike a go but it was too difficult to ride. Ex: Do you want to have a go at this game?