I have also considered leasing a small serviced-apartments building to the tune of $100,000 for a period of three years.

Is it recommended to allow for a seasoning period before selling the underlying property?

Essentially, this will ensure that your tenants are paying the same "amount" in rent as they have been the entire lease period.

And then during the exchange period they exchange cash from somewhere else for the note in the exchange account. 

Make the offer Once it is accepted you have to open escrow with a closing attorney.You proved the escrow attorney a copy of your contract along with an earnest money deposit.You should write in  an inspection period on the contract.

You will get a copy of the leases during your option period, and that's about it.If you are buying 5+ unit properties, then you have to be good at analyzing their financials, or work with an agent and lender that will take the time to work through those with you.

If not is there a inspection period that the tenant has to complete move in inspection?

I used that knowledge, and the commissions from my closings, to buy 24 units with my wife in a 18 month period which pushed us into "Lean FI". 

Once you have done multiple deals over shorter periods discussing reduced rates makes sense. 

Period.