[Nightedge]

Is it correct to say

-I suggest that our sales representative shall visit you this Friday.

Usually the verb in the that-clause following suggest/demand/request would be a bare-infinitive or one following should; but when can we use shall, which is found in some older works by famous writers?

[susan53]

No. The most usual version would be :
I suggest that our representative visits you....

The use of the infinitive (I suggest that our representative visit you...) would be very formal - I'm not sure why you say it is the "usual" form: have you checked with a concordancer? I can find very few examples where the infinitive is used rather than the simple present (obviously you can only tell the difference with a third person subject, as here).

If the suggestion was being reported so that second form verbs were being used then should might be used :
I suggested that our representative should visit them...

but I can think of no case where first form shall would be used after the first form suggest - certainly not in contemporary English, and it would sound odd to me in older versions too. I may be wrong there though. What are your examples and what period do they come from? If you can find some then there's the answer to your question : you can use shall after suggest if you're writing in the style of that century.

[Nightedge]

Excellent answer. Thank you.

[Nightedge]

Is my sentence correct?

-Just now Prof. Einstein suggested that we'd better postpone the lecture next Monday to a week later so that you can have more time to prepare for the mid-term exam and relax after that.

[susan53]

Yes, it's fine.

[Nightedge]

Good answer. Thank you.