I've smoked 1-2 cigarettes or 1 cigar a day for 6 years (not much!). Could I still get mouth cancer?

Yes. Even non-smokers can get mouth cancer and any use of tobacco increases the risk. Hpv infection is also a risk factor.
Yes. Yes, smoking 1 cigar a day for 6 years (6x365= 2190 cigars) could give you risk to have oral cancer. If you smoke cigars and also drink alcohol- the risk will be significantly higher. Please consider to quit smoking cigar and cigarette.
Possibly. Tobacco use (smoking, chewing, cigar, cigarette) all have an increased risk for the development of oral, throat, and lung cancer. This risk decreases after quitting. However, there is a strong association between amount of tobacco used per day and length of time used--so the sooner you stop the quicker your risk will return to a lower rate.