The Douay-Rheims is actually translating the Vulgate Latin, not Greek.
The YLT is translating a slightly different Greek text from the one you are using. Your text is consistent with the Nestle-Aland Critical Text:
αὐτῷ ἡ δόξα ἐν τῇ ἐκκλησίᾳ καὶ ἐν Χριστῷ Ἰησοῦ εἰς πάσας τὰς γενεὰς τοῦ αἰῶνος τῶν αἰώνων, ἀμήν.
The YLT is based on the Textus Receptus and the Majority Text, both of which omit the "καὶ" before ἐν Χριστῷ Ἰησοῦ:
αὐτῷ ἡ δόξα ἐν τῇ ἐκκλησίᾳ ἐν Χριστῷ Ἰησοῦ εἰς πάσας τὰς γενεὰς τοῦ αἰῶνος τῶν αἰώνων, ἀμήν.
The Greek Orthodox Patriarchal Text agrees with the TR and MT (which it usually, but not always, does). This may account for some of the gap you are seeing between your interpretation and the YLT.
εἰς πάσας τὰς γενεὰς
εἰς seems to mean just about anything, according to lexicons. The King James translates it variously as into, to, unto, in, for, on, that toward, for ever, and against. The Orthodox New Testament suggests to all the generations.
The phrase εἰς πάσας τὰς γενεὰς τοῦ αἰῶνος appears in the Septuagint, Tobit 1:4:
καὶ ὅτε ἤμην ἐν τῇ χώρᾳ μου ἐν τῇ γῇ Ισραηλ, νεωτέρου μου ὄντος, πᾶσα φυλὴ τοῦ Νεφθαλιμ τοῦ πατρός μου ἀπέστη ἀπὸ τοῦ οἴκου Ιεροσολύμων τῆς ἐκλεγείσης ἀπὸ πασῶν τῶν φυλῶν Ισραηλ εἰς τὸ θυσιάζειν πάσας τὰς φυλάς, καὶ ἡγιάσθη ὁ ναὸς τῆς κατασκηνώσεως τοῦ ὑψίστου καὶ ᾠκοδομήθη εἰς πάσας τὰς γενεὰς τοῦ αἰῶνος.
which Brenton translates as:
Now when I was in my own country, in the land of Israel, while I was still a young man, the whole tribe of Naphtali my forefather deserted the house of Jerusalem. This was the place which had been chosen from among all the tribes of Israel, where all the tribes should sacrifice and where the temple of the dwelling of the Most High was consecrated and established for all generations for ever.
τοῦ αἰῶνος τῶν αἰώνων
τοῦ αἰῶνος is in the genitive case and it serves here as a genitive of time.1 Other examples:
νηστεύω δὶς τοῦ σαββάτου, ἀποδεκατῶ πάντα ὅσα κτῶμαι
I fast twice a week [lit. "of the week"]; I give tithes of all that I possess. (NKJV)
οὗτος ἦλθε πρὸς αὐτὸν νυκτὸς καὶ εἶπεν αὐτῷ ...
This man came to Jesus by night [lit. "of night"] and said to Him ... (NKJV)
In addition to the observation about Tobit above, only other thing I can contribute here is that the phrase appears frequently at the end of Greek Orthodox Prayers, and it is appears in the English language version of the prayers as "unto ages of ages".
αὐτῷ ἡ δόξα ἐν τῇ ἐκκλησίᾳ καὶ ἐν Χριστῷ Ἰησοῦ
Regarding the doxology, omitting the καὶ leads to something similar to what is in the KJV:
to him - αὐτῷ
[be] the glory - ἡ δόξα
in the church - ἐν τῇ ἐκκλησίᾳ
in [i.e. "by"] Christ Jesus - ἐν Χριστῷ Ἰησοῦ
This reading actually seems to make more sense to me. Theophylact explained the passage:
The Apostle does not merely say, unto Him be glory, but adds by Christ Jesus. In truth, no man is able to give glory to God the Father, unless it is accomplished by the grace and power of Jesus Christ. He is the One Who gives us the grace to offer doxology to God, and He is the One Who teaches us how to do it. Aptly did he say that the glory of God is in the church: the church abides continuously, and the gates of hell shall not prevail against it.2 Thus, the doxology of God is also everlasting.3
Chrysostom seems to have had the other version of the text on hand (i.e. what appears in the Critical Text). He explains:
Unto Him be the glory, he concludes, in the Church and in Christ Jesus, unto all generations forever and ever. Amen.
Well does he close the discourse with prayer and doxology; for right were it that He, who hath bestowed upon us such vast gifts, should be glorified and blessed, so that this is even a proper part of our amazement at His mercies, to give glory for the things advanced to us at God’s hands through Jesus Christ.
“The glory in the Church.” Well might he say this, forasmuch as the Church alone can last on to eternity.4
1. Wallace, Greek Grammar Beyond the Basics, pp.122-124
2. Matthew 16:18
3. Explanation of the Epistle to the Ephesians (tr. from the Greek, Chrysostom Press, 2013), pp.52-53.
4 Homily VII on Ephesians