Closing can be done very quickly if you are paying cash for the home, the seller agrees to your price and terms, and you don't require any inspections.
If you are obtaining a loan, it is wise to allow anywhere from 30-60 days to close (depending upon where you are in your loan approval process, and what types of inspections, appraisals, surveys and/or repairs are completed, if needed).
A real estate purchase is the largest financial transaction that most people make, and a knowledgeable Realtor is a valuable professional to have on your team during the home buying and selling process. Interview several, and select the one to work with that will explain the process to you, and educate you with all of the facts you'll require to make the best decision for YOUR personal situation.
If you are not currently working with an agent, I'd be glad for you to interview me for the job! Call me at 850-686-9919 or email me at RealtorBarbaraMurphy@gmail.com . You can also visit my website at TartanPropertiesFL.com to learn more about me and the Home Buying and Selling Process. :)
For the second part of your question about the amount of time a seller should be allowed to move out....simply stated, it's highly recommended that buyers not close on their transaction until the sellers have vacated the home. To close with them present could be to embrace a mountain of problems that otherwise could have been avoided.
Generally attorneys and agents are hesitant to extend occupancy for the seller beyond closing. It introduces complexity and uncertainty for the buyer. What happens if you close, the seller gets your cash and then refuses to leave? What if they leave but damage the property before they go? What if they encumber the property in some way? All of these can be avoided with possession at closing. That said, many deals are done with a post-closing possession date. Unless you anticipate a long-term leaseback arrangement, the length of time from closing to possession should be short.
Closing can be really quick for a cash sale to an eternity for a short sale. It really depends on the complexity of the sale and every one is different. Cash deal ~2 weeks, conventional loan ~45 days, short sale ~who knows.