「さっきから七夕に関係ねェもンばっかり吊りさがってやがるなぁ」
These days they only hang things which are unrelated to Tanabata.
These days, on Tanabata, they only hang unrelated things.
I would have expected the particle と to be used in place of に for my first translation attempt, which is why I also gave the second attempt.
Which translation is correct? Is there any ambiguity? Can 関係 take both と and に? Is there a difference in nuance?
「さっきから七夕{たなばた}に関係{かんけい}ねェもンばっかり吊{つ}りさがってやがるなぁ」
These days they only hang things which are unrelated to Tanabata.
These days, on Tanabata, they only hang unrelated things.
First off, both 「～～に関係ある/ない」 and 「～～と関係ある/ない」 are correct and used equally often. To say there is a difference in nuance would be nitpicking. Thus, in the sentence above, the 「に」 can be replaced by a 「と」 without changing its meaning the least bit.
Exceptions: When talking about having a man-woman relationship, only 「(person)と関係がある」 can be used.
Moving on to your translations..
Your first translation is good though it is a free TL. The original does not say "these days" or "they". The subject of the original sentence is 「もン」, which refers to the decorations themselves, not the people who hang them.
「吊りさがる」 is an intransitive verb. Things are the subject.
「吊りさげる」 is a transitive verb. People are the subject.
But your first TL certainly captures nicely the essence of the original.
Your second TL is way off, I am afraid. The original does not say "on Tanabata".
My own TL:
"For quite a while, (I've been noticing that) mostly things that are unrelated to Tanabata are hanging (here)."
例文に限ると、「に」と「と」は日常あまり区別されずに使われており、相互に入れ換え可能です。
In the given sentence, usually,「に」 and 「と」 are used without being distinguished too much and are replaceable mutually.
しかし、ニュアンスとして、「に」の方が「と」に比べて、「関係する」ことに関して方向性と執着性が強いような気がします。
However, as a nuance, "に" seems stronger in directionality and relationship than "に" between the two related things.
「さっきから七夕に関係ねェ…」でニュアンスの違いを入れて考えると次のように説明できます。
When I put the difference in nuance in a given sentence 「さっきから七夕に関係ねェ…」, I can explain it as follows;
七夕｛たなばた｝「に」関係｛かんけい｝しないものは別｛べつ｝の何｛なに｝か「に」関係｛かんけい｝する、あるいは別｛べつ｝の何｛なに｝か「と」関係｛かんけい｝する。
The thing which is not related to the Star Festival by the particle に is certainly related to something else.
七夕｛たなばた｝「と」関係｛かんけい｝しないものは別｛べつ｝の何｛なに｝かと関係｛かんけい｝するかどうか分｛わ｝からない。
The thing which is not related with the Star Festival by the particle と is uncertain whether it is related with something else.
「に」と「と」の違いを図で示します。
I'll show you the difference by the illustration shown below.
