May Viably Be Construed as Either Middle or Passive Voice
Your observation about the grammar of the verb compared to the English translations is very astute. Unfortunately, I do not think grammar itself will entirely answer this question. And context fits either option as being viable.
If Passive, then in a Sense, All Three of Your Ideas are Correct
The topic of the passage here is in v.8, being "the word of faith" (τὸ ῥῆμα τῆς πίστεως), which itself is the means of righteousness by faith (v.6-7) not by law (v.5). Note that in v.9, the ὅτι clause is clarifying the content of this "word of faith which we preach."1 That is, Paul and those proclaiming the gospel preach the message of v.9, which is a personally directed call for individuals ("you") to believe and confess the work of God in Jesus in order to be saved.
So the conceptual subject of "the word of faith" is set up for in v.5-7, named in v.8, and more specifically clarified (quoted/paraphrased) in v.9.
So if the passive voice is intended, the subject of v.10 is what was named in v.8 and expanded on in v.9 (NKJV with two additions in brackets):
the word of faith = if you confess with your mouth the Lord Jesus
(which we preach) and [you] believe in your heart that God has raised Him from the dead,
[then] you will be saved.
The entire concept (named as "the word of faith") is what is being referenced by the expansion of v.9. But v.10 is also expanding upon the concept, for it is reiterating the means already noted in v.9 (heart for belief; mouth for confession), but v.10 is linking specifically belief as resulting in righteousness (thus continuing to expand upon the meaning of v.6, "the righteousness of faith"), which true belief should manifest in having the same confession about Jesus that they are preaching, and if so, one is guaranteed the result of salvation.2
So because "the word of faith" is the topic, it includes the concepts of all three of your possibilities, because all of v.9 is contained in "the word of faith" named in v.8.3 The "it" of the verbs in v.10 then refer back to this concept.
If Middle, it Justifies English Translations' General Person, Active Idea
The immediate topical context of the act of believing finishes with v.11-13, in which is found a generic "ὁ πιστεύων ἐπʼ αὐτῷ" (the ones believing upon him, v.11), and "πάντας τοὺς ἐπικαλουμένους αὐτόν" (all the ones calling upon him, v.12), and v.13:
Πᾶς γὰρ ὃς ἂν ἐπικαλέσηται τὸ ὄνομα κυρίου σωθήσεται.
Each, therefore, whosoever might call upon the name of the Lord, will be saved
So the generic ideas found in v.11-13 are no doubt what prompt the English translators to make v.10 have a subject generically stated of "man" (KJV), "one" (ESV/NKJV), or "a person" (NASB).4 It considers the 3rd person singular contained in the verbs of v.10 either in a proleptic sense,5 where the later verses clarify the subject, or simply generically of themselves in that the "he" subject contained in the verbs, being in third person, is to be understood as anyone (generically) that does these acts of believing and confessing.
Making the subject personal with the middle voice of course does then demand a switch of the passive voice to an active voice for translation, from "is believed" to "believes" and "is confessed" to "confesses." This switch is only clearly viable for translation on the basis of taking it as a middle voice (even if parsing information then incorrectly indicates only the possibility of a passive voice parsing).
This switch to an active translation based on a middle voice is okay, as it could be a middle voice idea because:
- In the context of v.8-13, it does not do any violence to the idea being expressed (since v.9 is fairly clear on its own what is needing to be believed/confessed by those being preached to).
- Certainly there is contained within the results of the believing/confessing that it would be for the benefit of the one doing so (salvation is the result), therefore the concept inherent in the middle voice is clear in the passage.
- It would fit either the indirect middle or causative middle ideas for the use of the middle voice.6
So while parsing of the words is regularly given as passive, the fact that they are present tense verbs does leave that parsing ambiguous and thus able to be grammatically considered a middle voice. A few online resources parse the ambiguity correctly (not making an interpretive choice of displaying only as a passive).7 So there is justification for the English translations being as they are.
So technically, a literal rendering (1) sticking with the passive voice in the English, (2) but not reinserting the whole content of v.9 into the mix, yet (3) isolating the key idea being referred to in v.10 could be (my translation):
For with the heart [the word of faith we preach] is believed unto righteousness, and with the mouth [the word of faith we preach] is confessed unto salvation.
That word of faith being believed and confessed consists of the concepts in v.9 (with v.12-13 also), which refer to the person/identity (recall "name," v.13) of Jesus Christ and the work demonstrated to be done by God's resurrecting Him from the dead, and trusting the same person by calling upon Him (v.12).
However, one cannot rule out (grammatically) the possibility of the middle voice idea, and thus justify all the English translations (except NIV, see note 4 below) putting a proleptic subject into the verse.
There is enough reiteration of the ideas in v.8-13 that either a passive voice or middle voice translation does not remove either (1) the concept that the word of faith is critical to be believed and confessed (emphasized by passive voice translation), or (2) that the one's so believing will be benefited thereby (emphasized in a middle voice translation)—both concepts remain intact for the whole passage with either type of translation.
1 For the use of a ὅτι clause to indicate reported speech (such as what was being preached here), see Daniel B. Wallace, Greek Grammar Beyond the Basics - Exegetical Syntax of the New Testament (Zondervan Publishing House and Galaxie Software, 1999), 456-458. I believe it is also possible to consider the clause as appositional or epexegetical (458-460), but since it is referring to the message of what is preached, all three amount to indicating the same thing, that v.9 is stating/renaming/clarifying "the word of faith which we preach." That it uses the 2nd person in the statement makes it more of an indirect quote idea, showing the content of what is preached, hence why I lean toward that.
2 One common use of εἰς is for showing result (Wallace, 369). That result is the intention is seen from v.9, as the result of the call to believe and confess was salvation. So v.10 is just looking deeper into how salvation comes, by gaining the righteousness that is by faith (v.6), not by law (v.5; i.e. not by "works," cf. Rom ch. 4)
3 (NOTE: reference to note #3 in the comments below is actually to what is now note #4.) It is worth noting that the entry on πιστεύω in William Arndt, Frederick W. Danker, and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 2000) supports more your #1 parallelism, stating (emphasis added showing support of that parallel):
Pass. καρδίᾳ πιστεύεται with (or in) the heart men believe (it=that
Jesus was raised fr[om] the dead) Ro 10:10.
4 (NOTE: this was note #3 when referenced in comments below.) The NIV oddly carries the 2nd person subject of v.9 into v.10, saying "your" and "you" in v.10. This is incorrect, as it not only changes the voice (which may be justifiable) but the person of the verb as well; plus it makes v.10 specific, when it is indeed a general reference.
5 Wallace, 318. Proleptic means the pronoun comes before the noun which it refers to, which is the postcedent of the pronoun (normally a pronoun comes after the noun, which is then antecedent of the pronoun).
6 See Wallace for the middle ideas—as an indirect middle would be the subject acts for the benefit of self by means of belief from heart/confession from mouth for his/her salvation (419-423); a causative middle would have a similar idea as the indirect middle, but also include the fact that the subject is "the source behind the action done in his/her behalf" (423-425). Probably one's view on election would influence which of these ideas is taken.
7 Online resources that do show the parsing as Middle/Passive not just Passive (as of 6/19/2014) are (1) Biblehub.com, (2) Perseus (click on πιστεύεται). The few commentaries I have found so far that note anything at all about it simply state the "impersonal construction" of the verbs (generally taking them as passive; "it is believed/confessed"); for example, Marvin Richardson Vincent, Word Studies in the New Testament (New York: Charles Scribner’s Sons, 1887), 3:115; A.T. Robertson, Word Pictures in the New Testament (Nashville, TN: Broadman Press, 1933), Romans 10:10.