Mazda 2 hard shifting, anyone?

Mazda, Mazda 2/Touring
Hello everyone :p

Im just wondering if any of your third gen. Mazda 2 auto has hard shifting issue? I saw many similar posts on Mazda 3 or 6, but failed to find any post on Mazda 2 relating to this issue. I also gather from Wiki that third gen. Mazda 2s use 4F27E auto transmissions which are also used on Mazda 3 and 6. And that made me confused as how can the same transmission be 4-speed and 6 speed at the same time? Maybe Mazda took out few set of gears from the 6-speed transmission to make it 4-speed?

My senior is like this: 2012 Mazda 2 Auto, with 18k miles (yes, not been driven much). Occasionally hard shift from 1st to 2nd or 2nd to 3rd, but never 3rd to 4th. The hard shifting feels exactly like when you drive a manual car, at a stop, and rev the engine high and then release the clutch fast to do a fast start. The car can also shift smoothly, and by smoothly, I mean you can hardly feel the gear change apart from hearing the change of the engine sound. The most frustrating thing is that the hard shifting is not consistent, most of the times the car shift with a little shake (could be normal), but in some situation, the car jerked badly into gear, and in few occasions the gear changes were very smooth.
From my observation, firstly, the hard shifting is somewhat related to temperature. iI was more obvious during the winter (< 40 F) with the car cold, and in the worst cases the car surged forward at the end of the gear change which literally shook my head back and forth.
However, during current day (~65 75 F) the car might also occasionally jerk forward when changing gears (even when the car has been driving for hours and reached working temperature) which leads to my second observation that the hard shifting is somewhat rpm related as well. When the car was well in working temperature, if I gave light throttle, the car would often jerk in 1st to 2nd OR 2nd to 3rd gear change but not in both for example, if the 1st to 2nd gear change was smooth, the car would often thump into 3rd gear. Or if the car thump into 2nd gear, the 2nd to 3rd transition would often be smooth. If I gave heavy throttle, all the gear change would be a little jerky, but often just a little shake (which is fine in my mind).
Lastly, I noticed when I disconnect the battery for few hours and reconnected it, the car would shift very smooth for few drives, and then everything was back to what I described above.
Here is some other related information if that might help:
I brought the baby 1 year ago (April 2015) from an elder gentleman with only 7k miles on the car. From the day I brought it, Ive noticed a slight hard shift of the car when going from 1st to 2nd, or 2nd to 3rd gear. So I did the transmission oil filter change (with OEM part) and filled it with new OEM transmission oil. The old oil was still pink, and only slightly darker than the brand-new oil, and no large metal flaks were found in oil pan or the filter, and only little ferrite dust was accumulated on the magnetic. (FYI. I live in NY state and probably due to salt on road the iron bolts on the transmission oil pan actually welded with the aluminum tread on the transmission block, and all the bolts with the thin end exposed to the air took the aluminum thread with it when I removed the bolt. So I have to use time-sert to repair almost half of the thread on the aluminum block. Please be careful when dealing with the transmission oil pan bolts on your Mazda). Another thing I noticed is related to the idle: When the car is in working temperature, at a stop, in drive, the idle speed would be 500 rpm and the car shakes a little as the engine is about to stall. And then, if I shift it to N, the idle speed would go a hair over 500 rpm and the shake soften a little. But if the car hadnt reached the working temperature, the idle would be around 600 -700 rpm depending on the temperature with a smooth running engine.

Many thanks for your time, and please say anything you would like or share your experience with your Mazda 2 :p