That's a lot of plants for such a small space. If it were me, I'd put the melons and pumpkins somewhere else. That would free up some space to use double squares for tomatoes and peas. Determine which side faces the sun and put the tall plants away from the sun and the short plants towards it. Good luck, I know what you mean about too many critters. Maybe this will give you some protection.

The tallest plants should go to the north so they won't shade the shorter plants.

Here is a link that might be useful: What's Growing On?

Here's a photo of our 4x4 plots with PVC domes & bird netting.

Which direction is N S E & W on your gazebo plan above ?
Now is the time to be paying attention to where the sun falls on your garden during different times of the day, so you will be able to plan your garden for next year. This changes as the seasons progress from spring to fall. If you plant things like spinach and lettuce so they will be in the shade of taller plants or in the area that may be sunny in the morning but then shaded by trees later, you can extend their growing season.

How many of each plant are you wanting to grow ?

Some of your veggies are planted in the spring and others in the summer. For example, this means a square that had radishes in it could be replaced with eggplant as the weather warms. Here is just one of many planting guides for Nebraska ... http://www.gardenguides.com/96192-vegetable-planting-dates-nebraska.html

You want to plant tallest items and trellised items on the north side of your garden. If not notated as spring, all the rest of these are planted in late spring and grow/harvest over the summer ...

Tall: corn
Trellis: sugar snap (early spring), tomato (indeterminate varieties on trellis), cucumber, honeydew,
Trellis depending on size/variety grown, or these may fall in the "short" category: watermelon & pumpkin?
Medium: bell pepper, jalapeno pepper, zucchini, eggplant,
Short: Early spring: Green Onions, Radishes, Turnips, Spinach
Short: Mid spring: Broccoli, Carrots, Leaf Lettuce

Turns out my Gazebo is 9ft by 9ft square...so here is what I've come up with... The top of the photo is the North side. North of the structure will be:

Strawberry Towers
Potato in Garbage Can

In the Gazebo:
Watermelon, Cantelope, Acorn Squash, Butternut Squash, Pumpkin

Corn, Cucumber

Onion, Spinach, Lettuce

Turnips

Green Onions

White Radish

Red Radish, Carrots, Bell Pepper

Cherry Tomato, Eggplant, Japanese eggplant, Broccoli

Larger Tomato (havent decided on what yet), Zucchini, Sugar Snaps

The items not in squares are hanging baskets of Grape tomatoes, and "topsy turvy" hot peppers.

Will this layout work? If no, what should I move, and where?

The taller plants, the corn, tomatoes, and trellised plants should be on the north side. (Not all of these plants will be growing at the same time, however. Corn and tomatoes in summer, peas and broccoli and fall or spring plants.) The watermelon, which should be planted in the spring, can be on the south side. The vines can spill over the planting bed. You should also check with your local Master Gardeners organization. They should have lots of free information.
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Here is a link that might be useful: University of Nebraska Backyard Farmer

I plan on growing the melons vertically, using the "pantyhose hammock" method.