Is the alternator going out on my Mazda or does it have a different problem?

No if it was your alternator bad it would not have cranked the next morning. Sounds like a bad battery or loose connection. Stop by your local parts store and ask them to check your battery. Most stores check them free of charge. Thanks

NEVER unhook the battery when engine is running. If the alternator is bad and overcharging, current spike would fry your engine computer.

No, the car needs 12 volts to work, even the alt. Remove the cable and nothing is powered and dies. As said above, never do this again.

Roy

Carolinacarconnection is correct about the problem not being related to the alternator. At least the symptoms don't seem to point to that. But it is also possible as alternators can be working but the charging is insufficiet to get the battery fully charged and this can cause intermittent non starting due to a weak battery. Do as suggest, get the charging system tested correctly.

For vehicles of yesteryears you can disconnect the alternator to test but that is not the correct way. A voltmeter across the battery terminals would tell you if the charging is correct.

For modern vehicles, they are equipped with a lot of electronic components and power surge can fry any of the control units. They can be costly to replace.

Alternators can go out any time and sometimes they can be intermittent. Without proper testing techniques, you would not know.

You have been warned against unhooking the battery with engine running and if you wish to do that again, there is nothing we can do about it.