In addition to Mike's comments below, from a lending and underwriting perspective:
1.) It is very rare for any lender to accept sales comparables that have recorded sales dates older than 3 months and/or further than a mile or so from the subject property (distance exceptions can be made for rural dwellings and for unusual properties eg oceanfront).
2.) If you are qualifying for a Government insured mortgage (ie FHA, VA or USRDA) or a Conventional mortgage (ie one which the lender will resell to Fannie Mae or Freddie Mac), the lender is responsible to HUD (in the case of Government insured mortgages) or to the GSEs (in the case of Conventional mortgages) for the quality of the appraisal. The lender is not responsible to the borrower for the quality of the appraisal.
3.) Your post mentions three types of valuations: Value for mortgage financing (which is the relevant value), value for taxation (which is the value determined by the Tax Assessor and is recorded at town hall), and the value for resale purposes (which can be heavily influenced by items such as remodeled kitchen and baths).
Typically in lending, remodeled rooms do not add appreciably to value since such improvements cannot be determined to be common among the selected sales comparables. Tax value has no relation whatsoever to appraisal for lending purposes (you might consider using the lender's appraisal as a basis to reduce your Ad Valorem property tax).
4.) Under the new Home Value Code of Conduct in process of being implemented by Fannie Mae and Freddie Mac, banks, lenders, and mortgage brokers are no longer permitted to use staff appraisers or independent appraisers (motgage brokers are prohibited from ordering appraisals under any circumstance under HVCC).
The lender must order the appraisal through an appraisal management firm and must not permit contact beween the loan originator and the appraiser. Some lenders have already implemented HVCC in advance of Fannie's and Freddie's deadlines.
One of the drawbacks to HVCC is that lenders are discouraged from altering an appraiser's estimation of value unless the appraisal contains obvious faults (such as using single family residences to value a condominium). A second drawback is that borrowers must pay for a brand new appraisal if they switch lenders. (HVCC is the result of an agreement between the Attorney General of New York and Fannie Mae and Freddie Mac as a result of some 250,000 alleged fraudulently influenced appraisals in New York state ordered by Washington Mutual Bank's retail loan division).
Your options, unfortunately, are few... especially if the lender has already implemeted HVCC. You may request the lender to order a second appraisal (which will incur a new fee), request a review of the appraisal (also an additional fee), or withdraw your application in favor of a new lender (which will require a nw appraisal and fee). Under HVCC, some lenders may not permit you to speak with the appraiser as long as your loan application is active with the lender. Bear in mind that lender are under tremendous pressure not to play around with appraisal values from state and Federal regulators.
In the current lending environment, you might be better off ordering your own appraisal before proceeding further (such an appraisal CANNOT be used by the lender, however) or, after receiving a copy of your appraisal, paying for your own appraiser to review the work for possible errors.
I wish I had better news.