Are you allowed to start filling out your college application at the beginning of your junior year?

No set rule. When you submit it is regulated. When you start to fill it out is irrelevant. But really, you should be looking at the application in your junior year and figuring out everything you need in advance. Then the actual filling it out takes little time if you have already written down what you need to put in it.

Applications would not be open.

If you graduate summer 2019 and would start college fall of 2019, applications would open up August 2018.

Too soon. Would not be considered until they are accepting applications for the particular term you would enter after high school graduation.

Eh. No, not really. If you look on the Common App or on a university's specific application it will say that this is the application for Fall 2018 admission. If you fail to complete and submit it by the deadline (which is fast approaching for most universities) it will be considered void and you'll have to start over again. Applications for Fall 2019 will open in August of 2018. What you CAN do is look at the applications so that you can see what types of information you'll need to compile. Keep in mind that the essay prompts do change from year to year, so you can't necessarily write your essays now.