'と' is the particle for 'そうじをして'. It explains the content of 'I asked'. It is 'to clean'.

母に「そうじをして」とたのみました。

I asked my mother 'to clean'.

you can use like following :

I said my mother to clean. 母にそうじをしてといいました。（言いました）

とcan be used as a quotative particle, it denotes what you asked, while に is being used to denote who was asked.

No it's not necessary and should not be there

omg lol

母を掃除する = "{someone} cleans my mother" ! :D

I assume you meant に instead of を in your sentence? Although する is a plain verb form, して is also plain (doesn't have ます or any other politeness auxiliary).

The conjugations of する are:

1, せ or し or さ [negative stem]
2, し [infinitive stem]
3, する [declarative]
4, すれ [hypothetical stem]
5, せよ or しろ [imperative]

How these conjugations are used:

1: し+ない "don't X" [non-past, negative]
2: し+ます "do X" [non-past, polite]
2: し+て "do X" [conjunctive, plain]
2: し+た "did X" [past, plain]
3: する "do X" [non-past, plain]
4: すれ+ば "if you do X, then ..."
5: しろ "do X!" [imperative, plain]

It's し+て above that we are interested in. It is used with various auxiliary verbs:

して+いる "is doing X" [non-past, plain]
して+いく "go do X" [non-past, plain]
して+みる "try to do X" [non-past, plain]
して+あげる "do X for {someone}" [non-past, plain]
して+くれる "do X for me" [non-past, plain]
して+くれ "do X for me!" [imperative, plain]
して+ください "do X for me!" [imperative, polite]
...
But one of the ways it can be used on its own is basically the same as meaning "してくれ" just without explicitly saying the くれ:

して "do X (for me)" [light imperative]

Now, with all that background information out of the way...

What is the main verb of the sentence?
= 頼みました
= asked

Who was being asked?
= 母に
= (my) mother + に (indirect object particle)

What was the asker saying?
= 掃除してと
= "clean (for me)!" + と (quoting particle)

if I recall correctly そうじ is "the cleaning" while そうじする　is the verb, "to clean". One can think of it as "do the cleaning"

Adding する to a noun makes it a verb in many cases. It works in the same way that adding もの to a verb makes it into a noun.

そうじ is a noun, the act of cleaning, so I think that would be something like "I asked my mother cleaning."

I think it should be correct and simpler than the example sentence. Reported.