The root meaning of あまり is overdose (of something). Like something is too much or over the top limit.
It comes from 余る (surplus).
It can be used as a noun. For example:
苦痛のあまりに泣いた。 : She cried on an overdose of pain. (if it makes sense)
It can be used as an adjectival participle. For example:
値段が余りに高い。 : the price is too high. (it is overpriced, basically)
It can be used as an adverb (usually in a negative sentence). For example:
余り出来がよくない。 : He does not do very well.
It can be used as a suffix. For example;
百名余りの従業員。 : Employees over a hundred names. (more than 100 employees)
As you can see all of the usages basically are the same あまり and the meaning don't differ that much.
Now with the examples I think you can understand what に does, so I will only explain も:
も often brings a nuance of emphasis to what's behind it, as if it is something unexpected and/or a too much:
二回もやったよ。 : I even did it twice!
2時間もかかった。 : It even took 2 hours.
The も in あまりにも is doing nothing more than emphasis what comes behind it. You can see examples on this page:
When used to mean "so much... that", the only difference is the usage. It is not the actual meaning of あまり that you have a problem with here. You just have troubles understanding the particle に.
As it is mentioned on Japanesetest4you:
あまりに and あまり both mean “so much… that”, however the placement is different. あまりに is placed before verbs, adjectives and Nouns, not after.
Now if you have specific sentences you struggle with I can explain it better
I don't feel like there is a difference in meaning between two phrases like:
But, I just found a phrase in which replacing one by another would change the meaning:
I feel like the second one isn't quite right. I can't find an explanation to it but I'll edit my answer when I can think of something.
As for your other question, it really depends on the case, when あまり is used as an adverb, it cannot be after the verb.
If you use あまり after the verb in this case, it would not be an adverb but a noun. As for the meaning, it would basically be the same as I explained earlier.
Also, when saying "to much ... that", which extra nuance gives あまりに/あまりにも?
Here again it depends on the nature of あまり. If it is an adjectival participle for example like in 値段が余りに高い。, the に is part of あまり because あまり acts like an adjective.
But if it is a noun like in 嬉しさのあまり, the difference between it with に and without に basically comes down to its relation with the verb.
If you know the difference between something like するとき and するときに, you should have no problem understanding this. When に is used, it is directly linked to the verb of the following proposition.
Can you see the difference?