Anaerobic bacteria will generate either methane or hydrogen gas, so I would not expect any helium to be present in the samples. If oxygen is available, bacteria will be able to ferment any sugars in the sample to form carbon dioxide, which is more dense than air and would cause the balloons to sink rapidly.
Methane and hydrogen gas (and other gases) can be identified by an analytical method called gas chromatography. This requires special equipment and access to a laboratory. This type of gas analysis is commonly done by gastroenterologists in clinical laboratories. Here is a paper that describes this type of study: http://www.wjgnet.com/1007-9327/12/3088.asp
Do you have access to a laboratory that could analyze a few samples for you? If not, then you can describe this technique and explain what you would do if you had access to a laboratory.
If you are interested, I can explain more about the ideal gas law after you have obtained some results and have measurements for your balloons. Otherwise, since you haven't had chemistry, this type of analysis could be considered outside the scope of your project. However, it would be good to include some general information on this topic in your discussion section so the judges will know that you at least are aware of the subject.
This project requires really fresh manure. Do you have a source that you can collect the manure and set up your experiment quickly with a minimum exposure to oxygen? Also, anaerobic bacteria produce very aromatic volatile compounds when they grow, so it's best to set up this experiment outside of the house. And, temperature is important. The bacteria will grow faster at a higher temperature, so do try to set up this experiment when the weather is as warm as possible. You should have a thermometer and measure the temperature.
You have good questions. Let me know if you have any other questions.