Alternator Capacity for 1968 Dodge Dart 270/SL-6

(a) what is the capacity of the OEM alternator for my 1968 Dodge Dart 270 SL-6??. P/N is 2642537 per Chrysler Parts Catalog
You ask tricky question.
2642 537 is a 37 amp alternator with single groove pulley.
37 amp is Chrysler's rating. It's probably the maximum or close to maximum output at a standard voltage. There doesn't seem to be an industry standard so its up to the manufacturer how they want to rate them.
Output goes up with rpm. Change pulley diameter or run a little faster and maxium output goes up quickly from idle, where its very low, especially on the older alternators.
Chrysler's service test, done on the car, shows the standard (37 amp) alternator must produce 23 to 29 amps at 15 Volts, (measured at the the alternator) with the engine held at 1250 rpm and using a carbon pile for load and voltage reg.

(b) I replaced the above alternator w/ a 60A re-manufactured unit. Now when I start the car the AMP meter pegs to the RIGHT and slowly returns to the "middle". Battery voltage is 13.8 vdc and w/ lights ON, points in the middle (Good). Voltage regulator is the newer solid state type, well grounded to chassis.
Ammeter going right is what it should do after start. This shows current is flowing into the battery. Your ammeter may be labled C for Charge on the right. (different years and models are labeled different).
Once the battery is recharged, the meter returns to center - showing no current flowing to or from the battery.

Don't like the immediate "peg". I guess the starter motor is causing this. Lots of amps initially for start of engine, then, back to normal. Weird I think.
If it pegs. First try recharging the battery on a charger. The battery charge may be a lower than normal and so its drawing more current.
Voltage at 13.8 measured at the battery must be with the engine running. That's a little on the low side but within the specs (13.5 to 14.5 V at 70* F)
If bringing the rpm up a little increases voltage when the headlights are on, then its telling us the alternator is barely producing enough power to run the ignition and headlights at idle. Typical with the standard 37 amp alternator, more so if there's some oxidation in the connections.
However, probably worth checking for resistance (seen as voltage drops) before blaming the alternator.

Any idea why I am experiencing this? Does not happen w/ OEM alternator.
Above guesses are as good as mine. Even if the new alternator has much much better output at idle due to pulley or design, pegging isn't right. Mid range - like 20 amos would be the most normally should see after a start. Usually its more like 10 amps. So possibilities are the battery is not holding a great charge, battery is just really low in charge, resistance somewhere between the ammeter and the battery. I've seen this (pegging) a few times recently as well - using the Standard 'studaker' VR. recharging on a charger (this battery has seen some abuse) seems to have resolved it. Time will tell.