Should there be a comma after "Lee Hsien Yang"?

No, it seems fine.

No

Definitely No. That's three Nos.
Commas are used when you pause, for a very very short time. Full-stops are used for a complete stop - or break. Semicolons are used when you are pausing a bit longer... for example listing items.
The three full-stops (ellipses) I just used is for indefinite, or longer pauses.

Example 1
So whenever you are reading something (pause - comma) read it that you can hear yourself (pause - if you want to end the statement here, you would put a full-stop. However, if you are continuing the statement, use a comma).

So here we go again.
Whenever you are reading something, read it that you can hear yourself, and...
Notice the comma precedes and, because it's a continuous statement with a short pause. Never stop, and then start a new sentence with 'And'. That's bad grammar.

On the other hand...
Whenever you are reading something, read it that you can hear yourself.
Notice you use a full-stop, because you end the statement there.

Example 2
But to a disciplinary tribunal examining Mrs Lee's conduct, they changed their account (should be a slight pause here) and said Mr Lee Kuan Yew had given his instructions to Mr Lee Hsien Yang instead. It was Mr Lee Hsien Yang who had roped Mrs Lee in on Dec 16, 2013 (should be a slight pause here also) to help with the will, as he was travelling to Brisbane that very day, the couple added.

Rightfully a comma always precedes and, if you are continuing a statement with a pause, so hopefully a teacher did not write this.

I hope this is not too much for you.
Also, I notice you ended with 'the couple added', which indicates that you were quoting someone... but there are no quotation marks in your sentences. So that would be a problem.

I don't know what level of comprehension this is, but correctly, if I understand the paragraph, it should go like this...
But to a disciplinary tribunal examining Mrs. Lee's conduct, they changed their account, and said, "Mr. Lee Kuan Yew has given his instructions to Mr. Lee Hsien Yang instead. It was Mr. Lee Hsien Yang who had roped Mrs. Lee in on Dec 16, 2013, to help with the will, as he was travelling to Brisbane that very day." the couple added.

Always glad to help.
I hope I did not overwhelm you too much.
Feel free to PM me if you need further assistance.