First of all, I want to make sure that everyone understands that in both:
「~を問{と}わず」 and
「~がどうかは問題なく、どれにも同じことが言える。」,
the 「~」 part will always be a noun or noun phrase.
Both mean "regardless of (noun)" even though the latter obviously sounds explanatory.
Now, let us pick an actual noun to replace the "~" so that things will hopefully become clearer. The nouns often used in 「~を問わず」 in help-wanted ads, for instance, include 「年齢{ねんれい}」、「経験{けいけん}」、「学歴{がくれき}」, etc.
「年齢を問わずご応募{おうぼ}ください」 = "Please apply regardless of age."
I presume that you have no problem with that phrase.
「年齢を問わず」 means:
「年齢がどうか(=何歳{なんさい}か)は問題なく、どれ(=どんな年齢の人)にも同じことが言える」
Hope you are following me so far. That literally means "What the age (of the person) is is no problem, it (= the job applications) can be said about people of all ages."
- In the first part, 問題 is the subject of なく and が was omitted because it is such a common expression (as least I think that is true). If that is the case, what is the first が being used for? Why does it not require a verb?
I'll be honest; I see serious comprehension problems here.
The subject is not 「問題」; It is 「~がどうか」 and the predicate is 「問題なく」. Please remember that 「~がどうか」 functions as a noun. 「は」 is, of course, the topic marker.
The 「が」 cannot be omitted because the whole phrase 「~がどうか」 forms a noun phrase. The 「が」 is the subject marker within the little phrase 「~がどうか」 ("how (something) is").
「問題なく」 is the 連用形 of 「問題ない」. You do not need a verb to form a predicate, do you? 「この花は赤い。」 is a perfectly grammatical sentence without a single verb in it.
- Was it necessary to include は after どうか or could it have been understood (in the same was as が and を can often be understood in expressions like ~かどうか知らない and ~かどうか構わない).
The 「は」 is necessary because this is written Japanese, not informal spoken language. You should not compare this with 「~かどうか知らない」 or 「~かどうか構わない」 when those are the entire sentences.
The sentence in question is much longer with 「どれにも同じことが言える」 in the second half and the 連用形 in mid-sentence.