Is having a savings account worth it or not?

Banks no longer pay an interest rate that justifies having an account just to hoard money. I prefer an account with a brokerage firm that gives me the flexibility of having a checking account associated with it. 

Of course it's worth it. There is never a downside to having a source of money you aren't spending, that can be accessed easily. 

Financially, no; but if someone hacks into your checking account they will not get what is in your savings account ... and enough money in savings + checking can eliminate monthly bank fees. [Or go to a place with no monthly fees.]  Having money saved somewhere is always good.

They don't pay much in the way of interest but if it can keep you from spending on frivolous things they can be good.

You gotta save something.

No. Get an investment account or just leave all your money in the checking account until you can afford an investment account (though you don't need much)