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This sounds like citrus brown spot disease. Unfortunately, this is often caused by overwatering. The soil should be allowed to dry out quite a bit in between waterings. This will help control the disease. It can also occur from infected soil splashing up onto the tree, or fruits.

This article will give you more information: https://www.gardeningknowhow.com/edible/fruits/citrus/citrus-fruit-brown-rot.htm

What you describe may be "greasy spot disease".
http://edis.ifas.ufl.edu/hs263

Photos would help.

Or go through this list of possible citrus diseases from U Florida an see if you can determine a more likely disease. http://edis.ifas.ufl.edu/topic_citrus_pests_diseases

Also look under citrus pest management. Mealybugs are common, but they are usually white fussy, sticky masses, not grey.

You can remove the fruit, but they may not ripen off the tree and be usable if they are too immature.

Can you protect the entire tree with a covering? If not the foliage may get frost damage and set it back.

Yes water management is a key issue. Getting that right is the most important thing.
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm

From the photo, there appears to be some nutrient deficiency, indicated by the yellowing on some of the leaves. A slow release organic fertilizer specific for citrus will help, like this:
https://www.downtoearthfertilizer.com/products/blended_fertilizer/citrus-mix-6-3-3/

In Santa Rosa area you will need to be concerned about frost damage and protecting the tree when a hard freeze is predicted, which is possible December through mid March.

Here is an article with more information:
https://www.gardeningknowhow.com/edible/fruits/lemons/growing-lemon-trees-containers.htm

There are, actually, quite a few wild varieties that have the potential to burn skin when in contact. There are certainly many types of lemon that are not edible in any sense. Some are too acidic, and some, simply, have no usable pulp.

The vigorous shoots with large leaves may be the response from fertilization and/or the warm environment.

You can prune those shoots down half way, or to just above the older growth to avoid excessive height and top-heaviness. But if mine I would not remove the fragrant flowers. And from the flowers is where fruit develops. It appears you plant is happy and productive.