Is it coming in through rhizomes or through seeds? If it is solely at the edges, it is rhizomes, if you have the occasional blade in the middle, then it is seeds.

Obviously, the site had horrible soil. As you fertilized, you made it possible for grass to move in.

I'll try to answer your questions.
1. Grass uses nutrients and water that would otherwise be available for your vegetables, so yes, it does affect yield and size.
2. Both corn and potatoes need to be hilled. After they start growing, you should pull soil up around the plants. Potatoes form on the stem above the seed potato and hilling keeps them from becoming exposed to sunlight. Corn is a tall plant and puts out side roots that stabilize it in the soil. These roots need to be covered with soil also. So to answer your question, yes, 1 1/2, 2 ft. of soil is quite deep enough. I have no experience with watermelons.
3. Mulching heavily will discourage grass, as will frequent shallow cultivation (be careful around your corn and potatoes). Notice the "discourage"--not eradicate. We fight bermuda grass here, and it always creeps into our gardens. I just weed, mulch, and swear.

Yes. It did start around the edges of the box then moved in towards the center. While digging, I would dig up networks of the roots that grew under the garden. Often, I'd dig up these centralized clumps of underground roots (best I could describe them is like spider legs; pale yellow, segmented grass-roots that spread in all directions). This stuff even grew into hard clay that I had trouble digging through with my shovel.

I've also bought supplies to start a simple bucket drip-irrigation system. My thoughts are that targeted watering will mitigate unwanted plant growth around the garden.

Yep, you have grass that is all interconnected - like Zoysia or bermuda grass.

Unless you dig a good one foot down around the perimeter and insert some kind of aluminum/landscape edging that is deep, it will continue to get in.

I've seen the Zoysia grass roots in my yard go down six or more inches deep to get around the wood I sunk into the ground. Hopefully one foot is deep enough as I'm not even sure, but the stuff grows like weeds, spreads out, and takes over everything!

Here is a link that might be useful: BsnTech Gardening Blog

Yes, establish a perimeter with a barrier going down at least several inches, then proceed to kill all grass inside said perimeter.

My favorite method is to use a small paintbrush and to paint the grass blades with undiluted Roundup. Minimal waste and maximal kill, and minimal danger for any veggie that may be present (if you lose a drop and it drops on a vegetable, pull and discard the affected leaf right away).

The advantage of the method: a single brush stroke on a single blade kills all of the rhizome, or at least one foot of it, eliminating the problem once and for all. You have to do it when the grass is really vegetating, in May around here.

I have awful crab grass that invades my garden. Using solar mulch helped a lot last year: Solar Mulch at Johnny's Selected Seeds

I drilled small holes to allow water to enter the garden and cut out holes for the plants. I still had to mulch and weed around the plants as the grass grew in any opening it could find. But it was much easier to manage.

Rachel
GrafixMuse's Garden Spot

1: Any weed will take up nutrients and water that is meant for your plants and it will make the plants have to compete for root space and light. It's best to pull any weeds you can when they are small so the problem doesn't escalate.
2: Corn and potatoes should be fine in 1.5 feet. They should be hilled for best results. Without hilling, potatoes will be on the surface and exposed to the sun while corn will struggle to stay upright in the wind.
3: Pulling as much as you can out whenever you can. Mulching will help, but you'll need to pull most of them out before you lay a mulch down, unless it's a plastic barrier.