Thank you for posting your attempts!
These are some hard sentences to make sense of. A few users on this site might be able to answer with more precision than myself, but I'll answer with some dissection and maybe this will help come to a better conclusion. The big problem with lyrics is that, because they are meant to be poetic, sometimes they don't make all the sense in the world, and are up to interpretation, so at best you're getting my personal interpretation now.
With that being said, here's my attempt:
How might this town take back its barging in and its laughter again?
ずかずか is an onomatopoeia that typically is used in a rude, stormy, barging in sort of entrance. Also, 取り戻せる is in the potential form, which leads me to believe the どうして at the start of the sentence is asking not why but how.
I howled as though I was howling while wishing that I am able to laugh in the end.
This is a bit tricky even for me. It almost feels redundant a bit to say it like this, since 遠吠えする is 'to howl' but with the って (taken as a short ということ) as 遠吠えにしたということ, I get the sense of wanting to explain what the howl means, then it gets explained in the next part. 願って吠えてる is a compound verb 'howling and wishing', but 'wishing' is getting modified by 最後に笑えるよう, meaning something along the lines of "so that I am able to laugh in the end".
I'll put a ring on you that is inscribed with this moment.
The part that's complicated here is the part that comes before the に in this sentence. So, breaking this up a bit:
"This moment with the rut/furrow/groove/incision/etc." I find that 轍 is an interesting word in that the words for it in English can always mean something else. 轍 is referring to the track left behind by the passing of a wheel, or the furrow or groove made by a plow or something like that. Therefore, it can also be taken to mean an inscription, a plowing, some sort of incision onto the earth or other material, etc. 轍と共にある all modifies 今 which is "This moment" or "Now", and since it is being modified it is taken as a noun.
This is easier. "I want to put a ring on you". This could mean wanting to marry the person, or, else, entrap the person as if a ring were put on the whole of the person.
Hope this helps! The users who know better will probably tear my answer to shreds, though.