Appears to be powdery mildew to me. I have lots of problems w/ it here as well. Plants in this family are most susceptible, IMPE. There are resistant varieties, but this time of year, when it is both cool AND damp, it's often @ its worst, no matter what.

Immature fruits rotting & falling off may likely be due to lack of pollination.

You will probably have to hand pollinate at first light every morning when the flowers open up. Female flowers are stuck to a little fruit. The male flowers do not have a fruit under the flower. Take an artist paintbrush and collect pollen from the male flower orange center and brush the orange pollen onto the female flower orange center. The flowers will be closed again by 8 or 9 AM so it has to be done early morning. (you don't have to make the bee buzzing noise when brushing the pollen)

I haven't grown these as yet but they now have some varieties of parthenocarpic zucchini that grow fruit without pollinating. I will try them this spring.
Jung Seeds;
Perfect Pick Hybrid Zucchini (Jung 03561)
Just about the most perfect zucchini you can grow!
This superb summer squash has the unique feature of
being parthenocarpic, it doesn't require pollination
by insects to set fruit. As a result, it produces
high yields even when weather conditions aren't
favorable. It can even be grown under row covers or
in greenhouses successfully. The fruits are very high
quality with glossy dark green skin and nearly seedless
flesh that has excellent flavor. Optimum picking size
is when fruits are 6 to 8 inches long. The fruits in
our picture are nearly 12 inches long and still are
very high quality and nearly seedless.

Caveli Hybrid Zucchini (Jung 03526)
The light green partner to our popular Perfect Pick
Hybrid! Caveli is also parthenocarpic and has all
the great qualities of Perfect Pick, but has light
green, tender skin and nearly seedless white flesh
with outstanding flavor. The leaves of the bushy
plants are curiously mottled with light green, making
them almost ornamental. If you like Perfect Pick,
you'll love Caveli.

Also Burpee has;
Sure Thing Zucchini Hybrid (Burpee 54841A).
This Burpee zucchini bears fruit early in cool, cloudy
conditions even when there are no bees or male flowers
around. Its medium-size fruits are long and very tasty.
Strongest flavor, good with other vegetables. 48 days.

Lou

If you have a lot of male flowers, you don't even need a paintbrush (if you don't mind sacrificing them). I just clip off the male bloom, peel away the petals, and just use the anther as a brush to 'paint' the stigma in the female flower.

This year my zukes are *still* throwing off nothing but male blooms, which makes the parthenocarpic varieties seem like a very attractive option.

Alan

Hi Susie, those plants are gonners, soil has been allowed to remain too wet and they're losing the battle to Powdery Mildew, they won't produce anymore and I see holes in the leaves from the first picture so they also have worms/cats. Since this is the boy's project it's not a bad idea to let him experience this failure but to make the best of a bad situation once they see the plants are history tell them to pull the plants out, let the soil dry for 3-5 days and then replace them with broccoli sets from the Big Box, 4 plants for under $2 should turn failure into success as long as they don't keep over-watering... get them to back off on the water.

Tom

You have the beginnings of what WILL turn very bad quickly. It is powdry mildew for sure. Do not do nothing, it needs to be taken care of. I used ot get it real bad here this time of year, but if you use the neem oil as explained below as a preventative in the future (tons of good uses for neem oil) you will not have a probelm again.
Do the hand pollinating as other suggested. As for the mildew. Spray the heck out of the entire plant with Neem oil. Tops and bottoms of leaves until you see no more mildew. Then, sparay once a week with a good soaking as a preventative. Different people have different opinions on Neem, which is good. However this will work for you.

Neem oil has many beneficial uses also. Helps a lot with bad bugs. There a lot of posts here on Neem oil, if you want to learn more.