BDAG mentions many glosses for the Greek word ἀκοή, but none relate to a courtroom "hearing".
ἀκοή, ῆς, ἡ (in form ἀκουή as early as Hom.; freq., incl. ins, pap, LXX; PsSol 8:5; ApcSed; ApcMos 8; EpArist, Test12Patr, Philo, Joseph., Just.; Ath. [1, 2]).
① the faculty of hearing, hearing (Philo, Rer. Div. Her. 12 of images οἷς ὦτα μέν ἐστιν, ἀκοαὶ δʼ οὐκ ἔνεισιν; given by God Did., Gen. 162, 21) 1 Cor 12:17; but mng. 3 is also prob.
② the act of hearing, listening (Pla., Theaet. 142d λόγος ἄξιος ἀκοῆς; Antig. Car. 129 ἀκοῆς ἄξια; BGU 1080, 6; EpArist 142 w. ὅρασις; Jos., Ant. 8, 171; w. ὄψις 172) w. βλέμμα 2 Pt 2:8 (cp. New Docs 3, 61). ἀκοῇ ἀκούειν (Polyaenus, Exc. 55, 2; LXX) Mt 13:14; Ac 28:26 (both Is 6:9); B 9:2 (cp. Ex 15:26). εἰς ἀ. ὠτίου ὑπακούειν obey upon hearing with the ear, i.e. as soon as one hears B 9:1 (Ps 17:45; cp. 2 Km 22:45). ἀ. πίστεως hearing of faith (=that ‘hearing’ which Christians call faith) Gal 3:2, 5 (SWilliams, NTS 35, ’89, 82–93, but most prefer mng. 4b).
③ the organ w. which one hears, ear (Sappho et al.; POxy 129, 4; PGM 4, 306; 323; 2 Macc 15:39; EpArist 166) esp. pl. (Dio Chrys. 15 [32], 11; Aelian, VH 3, 1 p. 39, 21; oft. Philo; Jos., Ant. 8, 172; SibOr 4, 172; Just., D. 131, 4; Orig., C. Cels. 1, 48, 33) αἱ ἀκοαί Mk 7:35. εἰσφέρειν εἰς τὰς ἀ. bring to someone’s ears Ac 17:20 (cp. Soph., Ajax 147). εἰς τὰς ἀ. τινος in someone’s ears Lk 7:1. νωθρὸς (q.v.) ταῖς ἀ. Hb 5:11. κνήθεσθαι τὴν ἀ. have itching ears (i.e. they like to have them tickled) 2 Ti 4:3, cp. vs. 4. Fig. περιτέμνειν τὰς ἀ. circumcise the ears=make someone attentive B 9:4; 10:12.
④ that which is heard
ⓐ fame, report, rumor (Hom. et al.; Sb 7205, 8; 1 Km 2:24; 2 Km 13:30; 3 Km 2:28; 10:7) Mt 4:24; 14:1; 24:6; Mk 1:28; 13:7; 1 Cl 47:7.
ⓑ account, report, message (Thu. 1, 20, 1 ἀκοὴν δέχεσθαι of things recounted by others in the past; Just., D. 8, 4 ματαίαν ἀ. παραδεξάμενοι) πιστεύειν τῇ ἀ. (cp. Jos., C. Ap. 2, 14; Just., D. 8, 4 [Is 53:1]; Did., Gen. 218, 2) J 12:38; Ro 10:16f; 1 Cl 16:3 (all three Is 53:1). ἐξ ἀ. πίστεως as the result of a message (proclamation) which elicited (only) faith Gal 3:2, 5 (difft. Williams, 2 above). λόγος τῆς ἀκοῆς the word of proclamation (preaching) Hb 4:2. λόγος ἀκοῆς παρʼ ἡμῶν τοῦ θεοῦ the word of divine proclamation that goes out from us 1 Th 2:13 (RSchippers, NovT 8, ’66, 223–34 tradition).—AOepke, Die Missionspredigt d. Ap. Pls. 1920, 40ff.—DELG s.v. ἀκούω. M-M. TW. Sv. [Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). In A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 36). University of Chicago Press.]
As 4b explains, Romans 10:16-17 (and other passages) speak of a message preached which elicits faith. In the case of Romans 10:17, the preacher is God Himself (not the Bible). And certainly His words are the most trustworthy of all words.