After you have been extended an offer by email including the contract and after agreeing to the terms, how long does it typically take to physically sign it from both parties and make it official?
How long before signing a contract
In the US nowadays, you usually sign an offer via DocuSign and it happens immediately once everything is confirmed on both sides. In the other countries - once you visit the office.
The offer is only 100% complete after the background check which apparently can take around 4 weeks.
Depends company by company - most will give you a deadline for signing (although this can often by extended if you have a compelling enough reason).
If you are sure you want to sign, then no reason to delay. I would only delay if you are also waiting for the outcome of other interviews.
And as the Anonymous commenter says, the contract will generally have a clause stipulating the employment terms are contingent on passing a background check - so technically it is not fully official until this date.