Yes, it is a manual mode. Look at your owner's manual or Smart Start Guide, or go online to page 20 of this one: https://www.mymazda.com/MusaWeb/pdf/smar... to see how it works. Or search online for Mazda 6 owner manual

Not a car expert here, but the 3,000 mile oil change seems to have some detractors. http://www.edmunds.com/car-care/stop-cha... According to that article, once every 7,500 to 10,000 miles is fine so if you don't drive that much, once a year should work. It's always good to check the manufacturers recommendation for your make & model though.

It can be a few things. A bad strut, a bad sway bar bushing or end link, a bad ball joint, etc. Here is how you can check your suspension:
https://www.youtube.com/watch?v=FOk0tkGE...
Here is how you check for ball joints:
https://www.youtube.com/watch?v=62bVn7E8...
and here is how you change the sway bar end links
https://www.youtube.com/watch?v=YqOsW1X1...
If you have any questions, feel free to ask me!

You're creating a big vacuum leak by doing that which will cause stalling and/or rough idle.You can do a pcv system test though if you wanna make sure its working properly just for the he** of it.Remove the pcv valve from its mounted location but leave the upper hose attached.Let the engine idle while testing.Shake the pcv valve and make sure it rattles.Now place your finger over the bottom end of pcv valve and see if your finger gets sucked against the valve.If it rattled and sucked your finger against it,the pcv system is ok.You can also lay a piece of notebook paper over top of the oil filler neck(after you remove the cap)and the paper should get sucked against the filler neck.If it does,the pcv system is working ok.If not,the pcv hoses or system may be clogged or leaking.A failed test on the pcv valve as stated at the beginning of this post also indicates a clogged hose or system or bad pcv valve(if it didn't rattle)

I would start by either cleaning or replacing the I.A..C. on the throttle body. It controls the air allowed to pass by when starting the engine cold and coming to a stop. It is run by the computer and often when excess blow -by builds up does as you are complaining about. It can often be cleaned but you would need to read your repair manual on how to remove and replace it as all are different arrangements, I clean the with Throttle body cleaner and lots of patience working the crud out. the moving part must be made to b free moving when done. As I said they are all different in design. The code may clear after this.

When I ride my scooter, it seems to perform fine for about 10 minutes, then after a stop it will accelerate veryyy slowly up to a lower top speed and RPM than it should be at (like 20 MPH instead of 35+)

I've tried various things -- changed the spark plug (the old one was fouled), changed the fuel filter, even put some SeaFoam fuel cleaner in there (thought it might help if the carb was getting gummed up, other than just cleaning the carb)

So far none of this has remedied the problem. Is my engine overheating? Is my clutch or contra spring going out? Does my carburetor just need a good cleaning? Or is it something else completely?