You will be amazed the extra debt you can pay off from saving on CC fees. I too think that keeping it open until closing may be beneficial but it may not be beneficial either - I work for a CU and our loan officers look at your relationship/income/credit score etc and not focus too much on issues such as UTIL (I would recommend it to be low) and Average age - you can tell them that you closed it due to fees etc. I would say though keep it open until you close anyhow just to be safe.
If they consider the credit score they are impliciting considering UTIL. It is part and parcel of the score. Sometime the number of open TL's can be an issue. Too few is not a good idea.
There's no damage in closing the card before you purchase the home - unless losing that CL will affect your utilization percentage. Maybe get the calculator out - if your total utilization doesn't take a hit, you know you are free to lose the card and the monthly fees as soon as you like; it's done it's work in helping you rebuild - now it is just dead weight. (Actually, if your utilization does take a hit - pay your total CC balances down to better utilization percentages - then you can still lose the card without any pain). Good luck with your house hunting. We just refinanced at 4 7/8% Nice!!
It is always a good idea to have at least three open TL's reporting when app'ing for any type of financing.
Having at least one bank or major CC reporting is important. OP has 3 other CC's reporting -- if one is a major bank (or Cap One, Discover, AmEx), they're in good shape. One is all you need, and a bank overdraft will fit the bill if it's reporting to the CRA's.
And, of course, it's good to have installment accounts in your mix for the best FICO score.
Number of accounts becomes important when reporting balances - too many reporting balances can ding you. One CC reporting gives most people the best results - YMMV.