Maybe. If you have credit cards, student loans, or a car payment, it's better to pay those off instead, then start the 401k after that.
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Hell yes. The match is free money. You can roll
it over into your new jobs plan.
probably not since you won;t be there long enough to get vested in the matching funds - you can start an IRA
Yes, of course.
I strongly encourage to do so, it will be like you putting twice since they match. Do it.