Hi, Arti.

Both "It rained yesterday" and "It did rain yesterday" are correct.

In the past simple, the auxiliary "did" followed by the bare infinitive form of the main verb can be used in affirmative sentences to EXPRESS EMPHASIS. In speech, the auxiliary is stressed. 

Examples:

I did want to tell you, but I didn't know how.

He did arrive late.

She did go to a school, but she is not nationally certified. 

She did steal the money.

The same applies to the auxiliary "do".

DISCLAIMER: I am not a native English speaker.

I am only going to talk about common conversational usage.

"It rained" is a simple statement of fact.

"It did rain" is used to contradict an earlier statement. It is always a reply to a earlier statement. For example:

A: "It rained every day last week."

B: "It didn't rain Wednesday."

A: "No, it did rain on Wednesday."

B: "Well, it didn't rain in Newburyport on Wednesday."

A: "If you say so, but it did rain here in Scituate."

It can also when a single speaker at first isn't sure, and then says "I did" for emphasis: he's sure now.

"What's that sound? Is that thunder? Yes, I did hear thunder."

What's the difference between

Sometime last week, it rained

sometime last week, it did rain

There is no difference, but sentence 1 is more natural and common.

They are not correct the way you wrote them.

Last week sometime, it rained

Last week sometime, it did rain.

You can't use sometimes in these two sentences.

With a time after.

Sometimes it rains on Fridays

Sometime on Friday last week it rained.

Sometimes it x-s on/in/at/when/if/because

Sometimes it x-ed on/in/at/when/if/because

Jordi gave you the correct answer :)

Thank you!!!! :-)