What seems to have been missing from the explanation is that the MI rates for FHA loans have dropped. What this means is that any FHA lender now can offer you exactly the same loan you have with lower payment. It's a streamline refinance and it does not need any credit qualifying docs (appraisal, income, assets) and its based on the assumption that the situation and value of your property has remained unchanged since the closing.
1- The savings you get are not from "taking 28 years and spreading them through 30" as you put it. It is not as if your current mortgage balance is divided over 28 years. The savings are from the reduction in MI. How much are you saving? Approx. it is 40% of your current mortgage insurance payment. In addition if you have a rate of over 4% you will likely get a lower rate and save on that too.
2- While this is true you should not keep your FHA mortgage for 30 years regardless. Whether it is your existing FHA or the new FHA you need to refinance that long before your scheduled 20% equity. In other words just because you loan amortization states it will take you 10 years to get to 20% it does not mean that is true in real life. The increase in housing market could mean that you have sufficient equity to refinance without PMI in the next 2 years. So the calculations should be based on when you expect to achieve enough equity to convert to a conventional. Also this Streamline FHA loan should have absolutely no closing costs to you they should be paid by the lender, if they chose not to, go elsewhere. Most lenders would do this with all closing costs paid.
3) (You missed 3 and went straight to 4) Again someone did not explain things right to you. Depending on when you took out that initial FHA loan, part of that Upfront Mortgage Insurance you paid is refundable. So you should get some of those funds back, again depending on your initial closing date. Now in any refinance you skip a month's payment, meaning if you close in April, your next payment is in July. You could apply your June's payment towards the principal balance that is increased by this new Upfront Mortgage Insurance Premium.
4) (your 5) No they are not. Again that UMIP can not be paid by the lender, that is added to your loan amount, that is what's increasing the principal balance. But as I mentioned if this bothers you then you can use your June's payment to cover it. That $71 Quicken would have gladly covered it for you was it not for the regulations. QL can only pay for your closing costs and your prepaid escrow items (like all other lenders) however that $71 is neither of those things. In your payoff you will see that there is one month's Mortgage Insurance premium as a separate item. This is neither a closing costs, nor an escrow item. Therefore QL can not pay for it. Streamline regulations do not allow for your loan amount to be higher than your current unpaid principal balance + your UMIP, therefore this fee can not be included in the loan either.
In conclusion, what you are being offered is a FHA streamline refinance, which is an excellent product, costs no money to you, takes no effort, minimal documentation, and is definitely as described. Since all lenders absolutely love this product and would offer an arm to have you as a client what I would focus more than anything would be the servicing portion. In other words where my loan ends up after closing. Chase, Wells Fargo have excellent servicing. QL not so much. Small lenders will sell your loan and you do not know where it will end up. All of them will offer you the same things more or less since the product is so good they have to be competitive.