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It may have a fungus. This could cause the leaves to brown and die. You can spray the shrub with a fungicide to treat this possibility.

It may also be getting too much water. It is a plant that likes it drier than wetter. If the soil is not drying out enough between watering, this can also cause brown leaves.

It probably has to do with moisture and the way the plants handle dry environment, shutting down their processes during the hottest part of the day and the night -- that's when they look wilted -- then opening stomata, and moving moisture stored in their roots during the night into the plant tissues when the sun is shining.

There are several varieties of Mexican Sunflower and some are hybrids.
If your plant is a hybrid the seeds may differ from the parent plant.

https://www.gardeningknowhow.com/ornamental/flowers/mexican-sunflower/planting-mexican-sunflowers.htm

You should definitely harvest the flowers right after they die. They will usually be eaten by animals and won't have a chance to re-grow. Collect these and replant next year.

Here is an article for more information: https://www.gardeningknowhow.com/ornamental/flowers/mexican-sunflower/planting-mexican-sunflowers.htm

They won't continue to mature off of the stem, but I would say that your best bet is to harvest them now, and let them dry. You may get some seeds, or you may not. I fear the frost will destroy them, so it may be best to go ahead and try to salvage some.

They can handle a good pruning, so this will cause no harm. Regular deadheading will keep it healthy. You can cut the vine by about half, and it will not affect negatively in any way.

This article will give you more information about the plant (even though it is for the shorter cultivars the care will be the same): https://www.gardeningknowhow.com/ornamental/flowers/mexican-sunflower/planting-mexican-sunflowers.htm