1, the transmission does down shift or "engine brake" when you let off and apply the brakes to assist with braking, mine in my 11 does this and it is kind of odd feeling in an auto but nothing im complaining about only does it below 35ish mph up to 4k is too high it may need some minor servicing, I would start with a trans filter change and a fluid exchange(not flush)2-3 fluid exchanges will get 95% or more of the old fluid out a flush may do more damage so just better to avoid this. 37-38mpg? is it skyactiv? if so that's normal and if not then that's amazing. 22500 miles IS NOT HIGH MILEAGE high mileage is 150,000+ these days. There is no way with our shifter set up to start in second if you have shut the car off, just a deal you have to deal with on sporty type cars with manually shift able autos. dos it still have stock tires? if so they are past their service life and need replacement due to age LRR tires may help mpg a little but you will notice a difference in handling and grip with LRR tires so I would live with 37mpg and get grippy tires I currently have cooper rs3-a and they are fantastic so far, even improved my mpg slightly over the less grippy and loud firestones on it. (I have a non skyactiv 2.5L so the worst non speed 3 mpg but average about 26-28 before tire change in the 1500 miles since have averaged 27-29) the coopers are light which helps with mpg more than LRR does. the OE tires on these cars are iron weights and will create mpg lowering issues. lighter rims will help this as well. the general rule is for every pound you take off of unsprung weight (wheels tires anything not in the body) you effectively cut 100 pounds off the cars weight, and I can say that you can feel it, the coopers where 2.5 pounds each lighter then the previous firestones so my car lost 10 pounds of unsprung weight the handling is better (though grippy tires with stiff sidewalls may help this more as the coopers have) throwing the weight into turns is better the acceleration and braking feel better and my mpg has gone up.
on my non skyactiv I do about 1500-2000 miles a month and do 7500 mile oil changes main thing with that interval is getting a good filter, use 0w20 SYNTHETIC and a long life filter like fram synthetic(gold box) or wix xp/napa platinum etc. to last that long but a 7500 mile change interval is the norm these days, just always better off getting the best you can. spend a few more $ on the filter and oil now or a lot more on an engine later. now skyactiv maybe different but my dads 14 6 with skyactive says 7500-10000 miles for oil change on 0w20 synthetic my 11 3 2.5 says 5000-7500 on syn blend (I use synthetic and go the longer interval as it protects better, for longer then a blend can) Mobil 1 or Castrol edge are great choices for oil but honestly as long as you get a reputable brand and a good filter you should be ok.
with the cruise and shifting it depends on the speed and gear your in when it needs to shift for going faster, in my 2.5 it is actually most efficient in 5th gear(highest in my model) between 70-75 and will downshift to speed up if I am in cruise below that and it is in 5th(as it tries to do in regular auto mode above 40ish mph), I use the manual mode most of the time though as I like manual cars and like to control my shifting. 4th is most efficient for me between 30-65 then I go to 5th
my shift pattern for my 2.5 if I do it myself is 1-2 at like 10-13mph (right after take off almost) then 2-3 around 19-25mph (depends if I am needing to go faster or not) 3-4 between 25-35 (again depending on the speed and traffic I am in) 4-5 between 45-65ish mph (again depending on traffic and road speed limit) I will go to 5th at closer to 45 if the speed limit is around 40-45 or flat if hilly I stay in 4th most of the time to keep from downshifting and 4th and 5th at that speed are practically the same efficiency wise for me so id rather stay in the lower gear to get more power more efficiently, and closer to 50-60 if 50 or more on the speed limit as the 2.5 is most efficient between 2.5k-3.2kish rpms I try to stay in that range and shift accordingly up or down yours maybe different and skyactiv autos are 6 speed so different ratios (you just need to try it in manual mode and see what it gets you) I cruise at 70-75mph on the highway as that's the most efficient spot for me to cruise in, if I need to go slower I watch the rpm and if needing to be closer to 60 I keep it in 4th as its more efficient in 4th at 60 then 5th because 5th chokes out some power and keeps it too low in the rev range 4 cylinder n/a (not turbo) engines like to be revved even the turbo ones do, so they are not as efficient at low rpms as a v6 or v8 (comparatively obviously a 4cylinder at 1500rpm should use less gas then a v6 at 1500rpm) etc. the torque in the 2.5 starts coming on noticeably at about 2k rpm and gets efficient at 2.5-3.2k so look at your engine specs and look at the rpm the max torque is the range right around that is usually your target for efficiency. the 2.5 has max torque at 3.5k rpm or so. so right below and to the max torque is usually most efficient, depending on driving style, situation, speed etc. just play with it in manual mode and you will soon find the efficiency range you/the car likes also the 5 speed on mine says it learns from manual mode and uses that in auto the 6 speed on the skyactiv may too so you may learn it something. and yes in auto mode it WILL downshift when needed for power in cruise control as it will turn off cruise control if you drop speed too fast. again ironically going faster may help this and may help your mpg I get closer to 26-27 average going 60-65 and closer to 27-29 going 70-75 almost 80.
HINT: its skyactiv if it has the badge on the back dead giveaway it says skyactiv on it. and 16 inch wheels (hatchbacks)
this is the link to the 2012 maintenance schedule table from the manual:2012 Mazda MAZDA3 Official Maintenance Schedule
click the heading below the one with the mazda logo it opens a pdf with the schedule in it.