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Prevention is the best route as well as early intervention when it does sneak up on you. There are organic as well as chemical controls but they work best before the infection has gone rampant.
https://www.gardeningknowhow.com/edible/vegetables/squash/powdery-mildew-in-squash.htm
https://www.gardeningknowhow.com/plant-problems/disease/powdery-mildew-homemade-and-organic-remedies.htm

They won't usually cross unless they are the same species, but that is not to say that it won't happen. It can, and does on occasion. Now, unless you plan to use the seeds from the harvest to plant your next garden, then any crossing will not even be noticed. The fruit will not take on a different form from receiving pollen. The next generation seed will take the new genetics between the two, so unless you plan on keeping those seeds from harvest for next year there is no need to worry about cross pollination.

https://www.gardeningknowhow.com/edible/vegetables/vgen/cross-pollination.htm

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/controlling-cross-pollination.htm

There are so many different varieties of pests and fungus, if both squash and cucumbers are mushy every year, I would suggest taking a leaf, stem, fruit and soil sample to your local extension service or a local nursery to get a definitive diagnosis. Here's a link for a collection of articles that may help:
https://www.gardeningknowhow.com/search?q=squash%20problem

Unless you, absolutely, have to water twice per day (i.e. a meter tells you that the soil is dry down to about 3 inches) I would say that it is remaining too moist without a dry out. This can cause poor nutrient uptake, or cause soil infections that will cause the soil to be way to acidic to support plant life, but allow bacteria to flourish.

I see, what looks like, the start of a magnesium deficiency. This can indicate an actual magnesium deficiency, or low soil pH.

First, I would recommend testing the pH and nutrient content:

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soil-ph-plants.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm

You can amend, accordingly, once you see what the soil's profile looks like.

Here are some more articles that will help:

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/adding-lime-to-soil.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/fixing-magnesium-deficiency.htm

https://www.gardeningknowhow.com/edible/vegetables/squash/tips-for-growing-squash.htm

https://www.gardeningknowhow.com/edible/vegetables/cucumber/tips-for-growing-cucumbers.htm

https://www.gardeningknowhow.com/edible/vegetables/okra/planting-okra.htm

Both types can be planted between May and June; winter varieties can be planted into late summer and fall.
Here's an article for you, and welcome to gardening:

https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-5/zone-5-vegetable-gardens.htm

https://www.gardeningknowhow.com/edible/vegetables/squash/tips-for-growing-squash.htm

Either way would harbor their own set of insects. They usually don't take much issue with growing along the ground, as this is how they would grow in nature. Mulch would, definitely, make it easier to maintain the weed population.

Another thought would be to grow them on trellis. I do this with beans, quite often, and it would work for squash, as well. Bamboo makes a nice, inexpensive trellis.

Here are some articles that will help:

https://www.gardeningknowhow.com/edible/vegetables/squash/growing-squash-on-trellises.htm

https://www.gardeningknowhow.com/edible/vegetables/squash/tips-for-growing-squash.htm