Is the purchase price of a shed counted as an addition or improvement when figuring the cost basis for the sale of my home?

It could be considered an improvement if you need to increase your cost basis to minimize a taxable gain.

Can I include the $50.00 fee I had to pay to get the permit for the shed to be placed on our property?

It depends.  If the shed was for business purposes, the permit may be deductible as business expense. Otherwise, it is not deductible on your income taxes.  A shed, when used for a business purpose, can be depreciated according to the Internal Revenue Service (IRS) guidelines. If the shed is only used for personal uses, it will not be tax-deductible.