What does なり mean and how is ように used in this sentence?

Question

化粧をすると、声が高く、大きくなり、相手の目をよく見て話すようになるそうです。

My understanding:

When you wear makeup, your voice becomes higher and stronger [the なり grammar I don't understand], and your partner [becomes able to do something related with looking and speaking, but I don't understand what].

Answer 1

What does なり mean?

なり is the sometimes called ます form or ます stem of the verb なる. It is sometimes called this way because you can obtain it by changing the verb to its ます form and then dropping ます:

なる ⟶ なります ⟶ なります ⟶ なり

Essentially, it's a more formal equivalent of the て form of a verb (see the explanation linked by @sundowner in her comment). Therefore, your sentence would mean the same than:

化粧をすると、声が高く、大きくなって、相手の目をよく見て話すようになるそうです。

Note that using this ます form instead of the て form makes your sentence feel more formal or bookish, and than they are not exactly the same even though in your sentence they would function in the same way. If you want to know more details about the differences between the ます form and the て form of a verb, I highly suggest you to read this: Do I have a good grasp on the basics of what the continuative form is?.

EDIT: see the comment below by @Chocolate, my analysis on the second part of the sentence (ように) is probably wrong.

how is ように used in this sentence?

ように has several uses, but IMHO the ように here means "as if" or "as though":

相手の目をよく見て話すようになる

You see well your partner's eyes, as if they [the eyes] became able to speak.

Finally, the そうです bit means that the sentence is hearsaying, i.e. you are paraphrasing what someone else has said.