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what year is tha Remington sportsman 58 serial 143766v
By the serial number and markings on the barrel.
Remington's website has a history section that you can look at.
Please check the link for the Manufacture Date codes details on the Remington Society of America.
50-500 USD depending on specifics
100-500 USD
Lookup date code on Remington Society of America here: http://remingtonsociety.com/rsa/questions/barrelcodes
You don't, you use the barrel code. If you don't have the original barrel you can try calling Remington.
On the bottom of the receiver, ahead of the carrier cutout, or on the left side of the receiver toward the bottom. sales@countrygunsmith.net
Date code stamped on left side of barrel, go to Remington Society of America for diagram and chart.
You can call Remington at 1-800-243-9700 and go thru the menu selection. One of the menu options is to request information on your Remington based off the serial number. You read off your serial number and the guy looks up in the database the year the gun was made and what the original barrel was for that gun (if gun had barrel options like shotguns do).
They only way to do it it is to comb through all the serial numbers of all the different types of shoutguns Remington has made the past 150 or so years. Remington.com has a history section in its Sportsman's Library that you might be able to use to narrow it down.