Could you check these sentences?

I want to employ a secretary who/that knows Russian.
I want to employ this secretary who/that knows Russian.

I don't think (that) Emilia loves him.
I think (that) Emilia loves him.

I don't think (that) my father has returned home.

My brother wants to live in an apartment that/which has three rooms.
My brother wants to live in an apartment with three rooms.
My brother lives in an apartment that/which has three rooms.
My brother lives in an apartment with three rooms.

Perhaps/Maybe, It will stop raining. (I'm sure.)
Perhaps/Maybe, It may will stop raining. (I'm not sure.)

Let's walk this path although it's raining.
Let's walk this way even though it's raining.
Let's walk this path even if it rains.

Could you check these sentences?

I want to employ a secretary who/that knows Russian.
I want to employ this secretary who/that knows Russian.

Who and that in this context are interchangeable, although there are some people that prefer strictly to use "who" when referring to people.

I don't think (that) Emilia loves him.
I think (that) Emilia loves him.

I don't think (that) my father has returned home.

These are all correct whether you use "that" or not.

My brother wants to live in an apartment that/which has three rooms.
My brother wants to live in an apartment with three rooms.
My brother lives in an apartment that/which has three rooms.
My brother lives in an apartment with three rooms.

Use "that" when what you are adding is necessary information. Use "which" on "optional" information. Source: http://www.quickanddirtytips.com/education/grammar/which-versus-that-0?page=all

Perhaps/Maybe it will stop raining. (I'm sure.)
Perhaps/Maybe it may stop raining. (I'm not sure.)

"Perhaps" and "maybe" are interchangeable here. However, I removed the use of the comma here.

Let's walk this path although it rains.
Let's walk this way even though it rains.
Let's walk this path even if it rains.

All of these sentences are correct.

Very, very good! :)