Is it actually normal to combine if and would in the same clause? If it is, then what difference would these two sentences have?

If you would just speak to him for me...
If you spoke to him for me...
The first sentence could have different meanings based on tone. It could express frustration in some way:
If you could just speak to him for me, we could get this thing finally settled!
Having "just" there is what gives the sentence emphasis. Of course the context here is sells it entirely. It's also far more common to hear it this way, regardless of the context.

As for the second sentence, it gives the feeling of a "what-if" situation.
If you spoke to him for me, then we might come to a conclusion.
However, it also sounds more demanding than the latter.