For example, if one's parent died on 1 Nissan, and was buried on 2 Nissan, what is the last day on which kaddish would be said for them?
Some base the Kadish for the soul on the date of death. Others say it goes from the date of the burial (See Pnei Baruch 34:9). The amount of time a soul should have the kadish is the 12 months of geihinom. The minhag is to retract this to 11 months so that it doesn't appear as if the son is assuming his parent needed geihinom (Rema YD 276:4). I have heard allowances for non-children to go for 12 months, and even for children to try to daven for the amud (not as a chiyuv), but saying only the kadish delegated to the shatz.
Ihe 12 month time frame is based on the date of death, it would end the date prior to the date of death (the next year), that would be the last day of Adar with kadish ending the last day of Shevat. A leap year would not change things- it still goes by 11 months. The last day, the yom hafsaka, allows the avel priority over other 11 month mourners. If you go by the date of burial, it would be one day later.
Actual mourning starts after the burial, so the 12 months of mourning would end the day prior to the kvurah date (the next year), in this case, the first of Nissan. This also goes by months, so 12 months in a leap year would be the first of adar 2. It is a custom to mourn on the 1st yahrtzeit as well.