You are right that keeping your teeth and gums clean (and therefore better dental health) is easier when your teeth are straighter. To know whether you are missing a tooth, the easiest way is to start counting your teeth from the middle onwards. As a rule of thumb, teeth often come in pairs; so if you have 7 on your right, you should have 7 on your left. However there are some instances that this may not be the case and therefore would be better looked at by an orthodontist.
Invisalign is a great option for you if you are concerned about how fixed braces may look in your mouth. The duration of treatment will definitely depend on a case to case basis (the average about 12-18months). Invisalign treatment is expensive because of the customisation to your teeth. Do ask the clinics for their installment plan so that it is easier on your pocket.
Good luck!
