Summary
The question centers on how a writer may describe, or assign different attributes to the same, or different persons. To illustrate, assume there are two different attributes, A1 and A2 and two different persons, P1 and P2. How does a writer convey A1 and A2 are both intended to describe only one person and how does a writer convey A1 or A2 is only intended to describe one person while the other only applies to the second?
In the second letter to the Thessalonians, Paul has composed three passages where this issue is found: verses 1:1-2, verse 1:12, and verse 2:16. Based on all three, it is reasonable to conclude Paul intends verse 12 to be understood as describing a single person.
2 Thessalonians 1:1-2
The terms God, Father, Lord, and Jesus Christ are used in the opening verses:
1 Paul, Silvanus, and Timothy, To the church of the Thessalonians in God our Father and the Lord Jesus Christ: 2 Grace to you and peace from God our Father and the Lord Jesus Christ (ESV)
1 Παῦλος καὶ Σιλουανὸς καὶ Τιμόθεος τῇ ἐκκλησίᾳ Θεσσαλονικέων ἐν θεῷ πατρὶ ἡμῶν καὶ κυρίῳ Ἰησοῦ Χριστῷ 2 χάρις ὑμῖν καὶ εἰρήνη ἀπὸ θεοῦ πατρὸς ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ (mGNT)
In both verses Paul identifies two different: God Father of us and Lord Jesus Christ. In both cases, Paul avoids the use of the article; he simply places attribute and person in apposition. In addition, he uses the conjunction καὶ to convey the two different are equally a part of the church (verse 1) and grace and peace (verse 2). In other words, the source of both the church and grace and peace, is both God our Father and Lord Jesus Christ. Moreover, since there is only one God Father and only one Lord Jesus Christ, the article is unnecessary, as is obvious from the actual text. The English addition of "the" is a translator's addition which simply adds emphasis.
2 Thessalonians 1:12
so that the name of our Lord Jesus may be glorified in you, and you in him, according to the grace of our God and the Lord Jesus Christ.
ὅπως ἐνδοξασθῇ τὸ ὄνομα τοῦ κυρίου ἡμῶν Ἰησοῦ ἐν ὑμῖν καὶ ὑμεῖς ἐν αὐτῷ κατὰ τὴν χάριν τοῦ θεοῦ ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ
Based upon the opening verses, it is difficult to justify understanding τοῦ θεοῦ ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ to be speaking of two different, the God of ours and Lord Jesus Christ. This is so, because, as is clear from the opening verse, the article introducing the phrase, is unnecessary if Paul is describing two different. The only purpose the article serves in verse 12 is to connect both God and Lord to one, Jesus Christ. Stated differently, if in fact Paul was referring to two different, God and Lord, the proper way to make that statement is θεοῦ ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ as was done in verse 2.
2 Thessalonians 2:16
In addition to the question raised in verse 12, a passage is found at the end of chapter two:
Now may our Lord Jesus Christ himself, and God our Father, who loved us and gave us eternal comfort and good hope through grace (2 Thessalonians 2:16)
αὐτὸς δὲ ὁ κύριος ἡμῶν Ἰησοῦς Χριστὸς καὶ ὁ θεὸς ὁ πατὴρ ἡμῶν ὁ ἀγαπήσας ἡμᾶς καὶ δοὺς παράκλησιν αἰωνίαν καὶ ἐλπίδα ἀγαθὴν ἐν χάριτι
Paul begins the passage using the article to indicate one: ὁ κύριος ἡμῶν Ἰησοῦς Χριστὸς. He continues by using the article to identify a second: καὶ ὁ θεὸς ὁ πατὴρ ἡμῶν. He reinforces the meaning by using a second article ὁ πατὴρ ἡμῶν. Furthermore, when verse 12 is compared within the entire letter, the case for understanding verses 12 as one, is stronger:
Verse 1:1 θεῷ πατρὶ ἡμῶν καὶ κυρίῳ Ἰησοῦ Χριστῷ
Verse 1:2 θεοῦ πατρὸς ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ
Verse 1:12 τοῦ θεοῦ ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ
Verse 2:16 ὁ κύριος ἡμῶν Ἰησοῦς Χριστὸς καὶ ὁ θεὸς ὁ πατὴρ ἡμῶν
Regardless of whether or not what is today called Sharp's Rule is valid, it is clear Paul's selective use of the article before and after καὶ, is following the basic sense of how a writer uses the article when composing passage to describe attributes of one or two different persons.
Conclusion
Obviously, Paul can distinguish two separate by avoiding the use of the article altogether; equally clear, he distinguishes two separate by using the article for both. Therefore, if he only places the article to introduce a passage, his purpose is to indicate two attributes describe a single person.