A VA appraisal stays on file with the VA (and therefore "sticks to the house") for a period of 6 months. This would only affect VA loans on your home. FHA also stays with the house for 6 months and only affects FHA buyers. I've had VA and FHA appraisals within two weeks of eachother that varied by $15,000 dollars. It is definitely not an exact science. Your buyer can challenge the appraisal and your listing agent can assist with that by providing comparable data disputing a low appraisal.
Unfortunately with new appraisal regulations it is possible to get an appraiser from a different county or state to assess a property. This is supposed to keep it fair and "arms length". However I have found that without knowledge of a particular area the comps can be greatly skewed either way.
Best of luck to you.