～ますまい is grammatical, but it's already fairly uncommon. It's mainly heard in role languages for stereotypical samurai and pompous/noble elderly people.
～ですまい is ungrammatical, although ～ではありますまい is okay.
Basically if you used ある for a person, it would sound more or less archaic.
～くありましょう is grammatical, but it's fairly uncommon and ...
In this sentence, the bolded part splits into two pieces:
「働いて｛はたらいて｝」(て-form of the verb 働く｛はたらく｝)
「います」(ます-form of the verb いる)
The て-form of a verb has many uses, and in this case can be thought of as applying a progressive aspect to the verb 働く (similar to "-ing" in English).
The grammar construction of 「て-form」＋「いる・います」...
They are irregular conjugations. Copying from Wikipedia:
There is a sound shift to using い for both the 連用形 (masu-form) as well for the imperative form. As Wikipedia mentions, using the "correct" form sounds very old ...
I'm assuming that 輝やいて is a typo for 輝いて. (Maybe it's a variant spelling, I don't know).
This て is simply 'and'. So 置かれて輝やいていた is nothing more mysterious than "was set and was shining".
You ask about whether one action modifies the other. Many times a clause in て-form can adverbially modify the following clause, e.g.
I know this question is a bit old, but since I came across this as a prominent Google hit when searching for help on the same thing, I figured I'd post a more complete answer for anybody else who comes along..
For context, the line in question appears to be from the "よつばと!" ("Yotsuba&!") manga, in the first volume, page 8 (3rd panel). Yotsuba and her ...