If I take courses while enlisted, am I eligible for G.I. Bill afterwards?

Those courses are paid for by tuition assistance. TA is free for you to use, nearly unlimited, and does not obligate you to apy any time to the Army. It's one of the single best military benefits.
There is a cap of 250 per semester hour and a maximum of 5 classes a year. I think that's the current cap, it's changed several times over the last few years.
In addition, with the college you're taking you will be eligible for a Pell grant when you submit your FAFSA. That will pay for another class that year. In addition there are several tax credits that will refund a portion of your books if they weren't covered by your school.
You can earn a certificate, associate, and a bachelor completely with TA and not having to touch your GI Bill. If you pursue a master degree, it will pay up to 250 per semester hour, the rest you will have to come out of pocket or use your GI Bill. Because most masters courses cost 500 or more per semester hour, you can use your GI Bill for what is called Top Up to pay the portion not covered by TA.

Yes, use TA ( tuition assistance) they have a cap per year that you can use also have a credit per hour cap. So research your school and make sure the tuition is under it. I recommend an online school that has a physical campus, look to some of the major universities , LSU, Alabama, Georgia etc and get your degree from there that way no one will ever know you got it online. You will have to get supervisor approval for TA, but that is generally pretty easy. I used it to get my masters when I was active. let me know if you have any questions.

DO NOT USE YOUR GI BILL- save that for something after

Yes, use your TA, go to the education center, they will hook you up. I recommend Post University if you go to online college.