You can get your AGI before your CFI as they are two different certificates. You need to take the Fundamentals of Instructing test along with the AGI, BGI, or IGI test and take both forms and an 8710 to a FSDO. I don't remember any balloon or helicopter questions.
First question is yes!
Second question, well all I can tell you about FAA tests is that when I did my helicopter ATP add on to my airpland ATP most of the questions were concerning a 727. Go figure, Very few helicopter questions. Study the whole book and you will do fine!
Yes you can get the AGI first (still waiting on my permanent cert from December...) all that is required is FOI and AGI test results and an 8710 (with only your signature on it).
Don't recall a single balloon or helicopter question on the test, with the exception of one regarding wake turbulance FROM a helicopter being similar to that from a large a/c.
As an aside, anyone actually have a BGI? Doesn't seem to be much point to that, as you are presumably already studying for the CFI test and the information is identical for the AGI (with all of 12 extra questions for the AGI test bank, mostly applying to privileges of the certificate). Plus, though the test is shorter in duration, its no cheaper...
You can get a ground instructor certificate before you get your CFI. I had a BGI. My instructor encouraged me to get my BGI after I earned my Private. Twenty years ago the BGI written was the exact same test as the Private written except there were 100 questions instead of 80. In other words, rememorize the ASA book and you're set. You do need to take the FOI written, which you'd have to do anyway for your CFI.
After you do all this, fill out an 8710 and take your Airman Written Test Reports to the FSDO. It'll issue you a temporary. With a BGI, you can instruct Private ground school and sign off students for the Private written. I'd suspect you can teach and recommend students for the Rec pilot written as well.
It used to be that if you had a BGI and took the IGI written, you could take that written test report, an 8710 and your BGI to FSDO and it would give you an Advanced and Instrument Ground Instructor certificate.
Another question about your AGI.............It is my understanding that once you get your CFI you can just fill out another 8710 and take it to your local FSDO and they will issue you an AGI. I never did it because as a CFI you are already qualified to teach ground.
Speedtree is correct. Although you can teach ground school and sign off people for writtens as a CFI, the Ground Instructor certificate and your CFI are mutually exclusive. You need to take the written(s) for your ground instructor certificate. You don't have to take the FOI twice.