Should I round my amounts to the nearest whole number or can I include decimal points?

You can enter decimals for some entries when the program allows you to do so,  but it is not normally necessary. You can round numbers up if you want to.

I've always rounded up when entering amounts.  But this year I noticed that TurboTax allowed cents on the W2 entry (or maybe I just didn't notice before).  I've never imported my documents, so I wonder if Turbo Tax imports the cents amounts, and then does the rounding further down the line.

Yes, you have always been able to enter the cents then TurboTax would do the rounding. If you look at your form 1040, even if you enter cents on your W-2, you will see that the form 1040 does not have any cents, only whole dollars.