④バイデン氏はこれまで8月1日までに決定したいとしていました。
“Up to now, Mister Biden has decided that he wants to make his decision no later than he first of August”
Vocabulary:
これまで: up to now;
⑥ ⓐ アメリカでは、ⓑこれまで女性が副大統領を務めたことはなく、ⓒ大統領が職務を遂行できなくなった場合に、ⓓ大統領を継ぐ立場の副大統領の候補にバイデン氏が誰を選ぶのか、ⓔその判断に注目が集まっています。
Thoughts:
アメリカでは: America is the topic of the sentence as well as where the action is taking place.
これまで女性が副大統領を務めたことはなく: up to now, there has never been a female vice-president;
職務: professional duties;
遂行: execution (+する);
場合: case;
Should I parse it this way?:
大統領が・職務を・遂行[できなくなった]場合・に
I’m not sure what your parsing intends to show.
In my mind it looks like this:
[大統領が・職務を・遂行できなくなった]場合
遂行する is a する-verb and the が and を particules are related to it. Then, [nounAがnounBをverb] is a clause qualifying the noun 場合
So “case” along with some attributive information about it. So it’s a case of:
大統領: president ⇦ 『が』subject;
職務: professional duties ⇦ 『を』direct object;
遂行: execution (する) verb.
So it’s a case where it was not possible for the president to execute professional duties? Is that it?
ⓓ大統領を継ぐ立場の副大統領の候補にバイデン氏が誰を選ぶのか
大統領【だいとうりょう】president; ⇦ 『を』direct object;
継ぐ【つぐ】to succeed;
立場【たちば】position;
候補【こうほ】candidate, contender;
『に』indirect object is my guess;
バイデン: mister Baiden; ⇦ subject;
誰【だれ】who;
選ぶ【えらぶ】to choose;
副大統領【ふくだいとうりょう】vice-president;
I’m not sure here…
The verb is “to choose” 選ぶ and subject “mister Baiden”, so a Baiden choses. The direct object being 大統領. So Baiden chooses “the president”? That doesn’t work.
継ぐ立場の副大統領の候補
“Successful position” 
“Vice President” “candidate”
継ぐ means "succeed’ in the sense of “come after.” So this is more like “Successor position”
Yes, it is a verb. Do you just mean why is the English not a verb too? Because I can’t think of a way to say it naturally with a verb and not adding other stuff as well. “Person-who-succeeds-position” or something?
The important thing was it’s not “successful.” That’s a different meaning of succeed.
Yes, it is a verb. Do you just mean why is the English not a verb too? Because I can’t think of a way to say it naturally with a verb and not adding other stuff as well. “Person-who-succeeds-position” or something?
Not sure what you mean.
It seems to me that 継ぐ is used as an adjective here, not as a verb, thus my confusion.
Successor position
My guess based on a very vague idea of the context of this article is that this says something along the lines of ‘who will Biden choose as candidate for vice-president, who will be in a position to succeed the president?’
(I blurred the first bit so you can focus on the stuff that explains what’s confusing you right now.)
Verbs can modify nouns to make relative clauses.
食べる人 person who eats
継ぐ人 person who succeeds / inherits
I assumed as such but I want to understand the grammar as opposed to understanding the sentence because of my prior knowledge. 
Ah ok, that’s where the 大統領『を』is coming from.
There are two 『を』direct objects in the phrase. While I understand the second one, the first one I’m still not 100% clear about.
バイデン氏が誰『を』選ぶのか
That’s clear enough.
I guess you could say that after looking at the particles in order to figure out the role each word plays, you then need to use proximity to guess which are related. It’s 大統領を継ぐ as a unit because they’re next to each other, and because 継ぐ is in front of 立場, meaning that it’s part of a phrase (relative clause) that modifies 立場.
Yes. And this […]場合は or […]場合に can be very often translated as “In the event of” / “In the case of”
(Side note just in case, in English “phrase” doesn’t mean sentence, it means groups of words, expression. It used to trip me up a lot when reading article on grammars full of concept like “noun phrase” which don’t look like sentences at all 
)
There is usually only one direct object marked by を in a clause, but there can be many in the entire sentence. Also, usually, especially in journalistic paper, any nounを or nounが almost always link to the very next verb in the sentence. It’s true for the entire article, I think.
大統領を継ぐ alone would be a sentence meaning “succeed the president” or “inherit the presidency”
Looking at it as that alone is fine, right?
It then modifies 立場.
Yes but who is the subject then?
In this case it’s the noun just after, 立場の副大統領の候補. The candidate to the position of vice-president.
Unfolded, it would be:
立場の副大統領の候補が大統領を継ぐ。 The candidate to the position of vice-president will succeed the president.
It’s like:
田中さんがごはんを作った。Tanake made the meal.
田中さんが作ったごはんはおいしい 。The meal that Tanaka made is good.
ごはんを作った田中さんは上手です。 Tanaka, who made the meal, is talented.
Uh… I might be misunderstanding, but I think it’s slightly more complicated than that. I think it’s
副大統領の候補が大統領を継ぐ立場（にある・に立つ）。Like you said, the ‘candidate to the position of vice-president’ is the subject of 継ぐ, but 立場 is the position in which that candidate is or the status which that candidate has.
If we wanted a slightly more expanded version in which each sentence only has one verb, we could say:
必要があれば、副大統領の候補が大統領を継ぐ。それは副大統領の候補の立場だ。
If there is a need, the candidate for vice president will succeed the president. That is the position of the candidate for vice president.
Yeah, I think I made a mistake, you are right.
Leebo also said the clause modifie 立場.
大統領を継ぐ立場 : the position of succeeding the president
Maybe the subject is unstated ? Like
(人が)大統領を継ぐ立場 : the position of (a person) succeeding the president.
(大統領を継ぐ立場)の副大統領の候補
“The vice-president nominate in the position of (the person) succeeding the president”, Maybe ? The longer I look at this grammar, the less sure I am
It’s a bit frustrating, because it look like an English translation would just drop this “position” thing entirely and say something like “the vice-president nominate who will succeed the president if the president can not longer perform his duties”