Rather than dick about for ages just call out the AA or RAC and get them to test the battery. They carry midtronics battery testers which will tell you in seconds if it's duff or not. Failing that most garages have battery testers.

As for the battery causing the alternator to overheat I very much doubt it. It's either the alternator is goosed or you have volt drop on either the engine to chassis earth or the alternator to battery live cable (volt drop is essentially a resistance in the circuit which causes heat build up if severe enough)

Grab your volt meter and set it to dc volts.
Start the car and check the alternator is charging
Now connect the multimeter to the positive terminal of the battery and the positive terminal of the alternator.
You should get a reading of 0.1 volts or less, anything more is a resistance
Now connect to the earth on the battery and the alternator body and check the reading which should be the same as mentioned previously.

If you get any readings higher than stated clean all connections in that circuit including the battery terminals and clamps and test again. If the reading is still there than replace the cable as it is faulty.

Point to note the circuit you are testing has to be under load to test for volt drop. It can be used to localise wiring faults on any electrical circuit.

Laymans term for volt drop is usually bad earth etc which is caused by resistance and as such electricity will take the path of least resistance which is why when you have a bad earth on say the brake lights the indicators or such illuminate when you press the brake.

If you want to test the battery yourself with just a voltmeter then follow this procedure.

Firstly run the car for 5 to 10 mins
Switch off and check battery voltage it should be between 12.5 and 12.8 volts
If it higher switch the headlights on for a few seconds and then re check the voltage is between the figures already mentioned.
Now immobilise the car from starting so that it will crank over but not start.
Position the voltmeter where you can see it
Crank the engine for 15 seconds
during the cranking if the voltage drops below 9 volts that's a fail
also during cranking the voltage will stabalise to say for example 10.5 volts, if it drops more than 0.3 of a volts once stable then that is also a fail (ie cranking 10.5 volts but tails down to 10.1 volts or less it's failed)

 

Most batteries these days are calcium/calcium or calcium/calcium,silver. A charger for a good old lead acid battery is not ideal. Modern batteries are the main reason alternators now have high tech regulators often controlled by engine ECU. Chargers for above batts are available but not cheap!

 

A dumb charger by my definition is one that will overcharge a battery if left on too long. There are varying types but essentially it revolves round fixing the peak voltage and/or current, so you need to keep an eye on it. An "intelligent" charger on the otherhand will detect the drop in current flow and then limit the voltage down to a safe 13.2 volts, where the battery can still draw current, but not in a manner that will overcharge it, allowing it to be left on charge almost indefinetely.

A charger with an ammeter, is usually a dumb charger, which only limits the maximum peak voltage. This can be anything from 14.4v to 16volts! (which is why its often recomened to disconnect the battery from the car prior to charging).

When it comes to the varying current on a ammeter, it gets tricky to explain what you see, because a battery when discharged (and not sulphated) will readily accept as much current as the charger can throw at it. The measured voltage at the charger will be quite low, and the current will be quite big, as the current demand of the battery is so much that the voltage provided by the charger drops as it simply can't supply what the battery demands, this is down to the charger's built-in protection circuitry that limits peak current to prevent itself from overheating (a good thing as too much current in one go is not healthy for the battery's longevity either).

When the battery becomes more and more charged, it doesn't readily accept as much current, as such the voltage rises and current flow into the battery starts to taper off. So when the battery's charge is at its peak, current flow will be much less than at the start but the voltage will be quite high.

This is nothing to do with the charger controlling or reducing current flow (barring its inability to provide more current than its rated at), it is more to do with the battery's change in resistance.

A fully charged battery should draw about 2% of its Ah current rating or less (e.g 1amp for a 50Ah battery). Its at that stage when the chager needs to be disconnected. If its in a quiet garage, you'll clearly hear the battery bubbling away - this usually occurs when the battery is about 90% charged (or the battery is duff).

With reference to Calcium batteries ("wet" type Sb/Ca or Ca/Ca): Typically, these need higher peak charge voltages than conventional lead acid (Sb/Sb) batteries. So some lead-acid battery chargers won't fully charge them. Ironically, its actually the more expensive "intelligent" chargers that fall foul with these types of battery (unless otherwise specified - read the manual ). Although they will be perfectly ok to get you up an running again if you ever become stuck with a flat battery.