Congrats and welcome to Singapore & Singapore Expat forum.
Did you ask same thing to your employer? If yes, then what did they advise? If not, then pls ask them. ( You have answered most of your queries by yourself, in addition you need to go though a formal medical checkup on your arrival (within first few days).
Secondly, IPA letter issued by MoM electronically. Why do you think employer has to sign on it? There is no need.
You may spend sometime to go through “visa” category, you can find all your queries are discussed in numerous occasions. Good luck