Can someone please explain the need for the と before たのみました?
Perhaps the closest meaning of とin such sentences would be (that) or (to), depending on the sentence. Here (although I personally find this sentence a little odd), the meaning is (I asked my mother to clean up). An example for the use of と as (that) is 彼は来ると言った（かれはくるといった):He said that he would come.
I understand what you mean, but it is important to remember that in translation, literal translations of two sentences from different languages will not necessarily be identical (in vocabulary or grammar) to each other. So, although less common than "to" (and a bit strange in my opinion), you can think of the と頼みました sentence translation as (I asked my mother that she cleans up).
The sentence itself is fine and has no problems, grammatically. But a sentence where the child (regardless of age) was the one asking the mother to clean up made it seem a bit odd to me, as normally the opposite would be true. So, I wondered if the verb was actually 頼まれました (I was asked by mother to clean up). 頼まれました can replace 頼みました in that same sentence without the need for any changes, and make it a more common to come across in Japanese content. But, there is no structure problem in the sentence, and it may have really been written that way, because it all depends on the context out of which the sentence was taken from.
Thank you! I haven't yet studied the passive voice: adding まれ makes it passive, right?
In trying to learn more about using passive, I came across this Japan Times article which has useful examples:
It also taught me one more way to express (hopefully mild) annoyance:
Yes, 頼まれました is passive voice obtained by changing the last letter in the word from its basic うformat (頼む）to the あ format then adding れ（頼まれ). As for やられた (and as implied from the examples in the article you linked), I'm not sure that "annoyance" is the right category of what one feels when they use that word. You see, there is always a sense of "defeat" or "damage" in the sentences using やられた. I did some digging, and came back with some stories where the verb was used:
(1) 夜寝ている間に泥棒にやられました : in this story a burglar broke into the speaker's home while sleeping at night
(2)今年も害虫にやられたよ: a farmer complaining how this year too his crops were attacked by pests.
(3) 子供にやられた: a mother who discovered that her kids stole the marker pen and filled her furniture with graffiti
(4) 敵にやられた: someone killed by the enemy in a video game
(5)勉強はした。しかし面接にやられた: somone who studied well for the written test (during job hunt), but it was the interview that "got him"