Home financing through FHA requires a miniumum of two years from discharge and Fannie/Freddie are 4 years. That being said, you can get financing with just your wife on the loan if that qualifies you for enough purchasing power. Other than that you should look to your local Realtor community for someone who specializes in "seller financing" or "lease/purchase". If you happen to have 30% to 40% to put down, you may also qulify for a "hard money" loan throgh private/non-conventional entities
Two ways to look at this. Keep the loan balance (assuming $200K) the same, refinance at no cost for a 15 year amortization, saves you almost $140 per month. Savings after 14 years $23,520, BUT still have one year of payments to make, total of $17,453, for a net savings of $6,066. Or you can get the same loan but pay it on a 14 year amortization, saving $62 per month and a total of $10,438 after 14 years. The key in either scenario is that it may make sense IF you get the refinance with No Closing costs.
The answer to your first question is "Yes, a lender can require an appraisal on a cash out FHA strreamline". Though FHA does not require an appraisal your lender could be making a mistake on a couple of accounts. One, they may not have asked you if you have enough money in savings to pay for the closing costs. They would therefore need to include closing costs in the loan amount and since this increases the balance, an appraisal is then required. This may be the case if your loan amount is less than $150,000 and/or your current rate is less than 5.25%. The second reason is that they may not be that experienced with FHA loans and just don't know how to properly structure your refinance. In either case, it sounds like you should NOT pay for an appraisal and also look for another more experienced FHA lender.
Your answer lies in when you think your rate will adjust higher than existing fixed rates. The fact that you are in this loan currently suggests that you can handle some amount of risk in flucuating interest rates. Monthly LIBOR is likely to stay below 3% for 6 to 12 months, as well as fixed rates below 5%. If you feel you are going to keep this loan for more than 3 years you may want to go to a 5 year fixed ARM with rates currenty around 3%. As you feel you are likely to keep this property financed for over 7 years, it is likely a fixed rate in the 4% range will save you money over a 10 year plus time period.
Once we get a loan application from a borrower, as a Direct FHA lender, we are able to order the case number (can take until the next day) and then lock you in for the approriate time it takes to close a FHA streamline. This is the time from application until the end of the month with a minimum of 25 days and a maximum of 60. The lender you choose should be able to do the same.