I got one of those recently too.

However, I am wondering, why does it say that my first due payment is on 6/3/17?

Yes. The first statement is usually like that.

Can I pay some of the $199 i owe now, and will that lower the utilization?

It will lower the utilization, but it might not be reported.

I forget Chase's reporting (they report immediately if you pay the balance fully I think), but it is usually once a billing cycle or something.

Utilization is important, but in the beginning, it is not that important. Maxing out every billing cycle and paying it in full (not advisable!) will cause your credit scores to go up and down, but it has no long term effect (as of VantageScore 3.0 and FICO Score 8).

If you are going for rewards, it can be worth it to maximize that too.

Lastly, I have a 15 month 0% apr, then it goes to 24.24%, am I able to keep using credit (while staying below 30%) but making min payments for those 15 months without interest?

Check the terms. They should clearly disclose how that is handled. But for a $700 limit credit card, why would you carry a balance?

This is a rewards card. Try to forget the 0% APR and use it as it best benefits you: getting rewards.