Honda Odyssey jump starter B&D 300 amp.
It depends on the type of jump starter you have.
If your jump starter is using a lead acid battery then this is likely fine but you should refer to the manual.
If you’re using a li-ion jump starter then it will contain diodes to prevent current flowing back in to the li-ion battery as they will get damaged and explode from the charge off of an alternator.
This tear down of a li-ion jump pack’s cables on youtube by bigclivedotcom shows the diodes:
No you can not.
These small portable jump starters have a different type of battery than what your car has. you must use the included charger to recharge your jump starter.
leaving it connected may actually damage the jump charger as they are not designed to be charged fast, the way your car battery is.
Can you recharge jump starter by connecting to battery
You can, even at idle the alternator will raise the vehicle system voltage, causing the battery to charge, and the jump starter connected to it. This model has a lead-acid battery that will tolerate a fair rate of charging - obviously it's aimed solely at cranking duty, so they're more optimized for high currents rather than energy density. It's a sealed battery, with typically small rubber valves that vent the cells, so don't let them get overcharged (when they start gassing). The regulator on the alternator would prevent that, though.
However - never move the vehicle with those big clamps on the battery terminals. It's an unfused circuit with not very mechanically positive connections, and the potential for a lot of current to flow should they move.