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You can either cut them off at the base, or pull them out of the ground, to improve appearance in your garden. If you do not remove them they will simply turn into mulch and eventually work into your garden soil.

This is an interesting question and there are two possibilities for the answer.

If the seeds are more off-white or beige than pure white, this is probably an issue of different varieties. Different morning glories can have different colors of seeds. Since you have multiple flower colors in the same garden, a few varieties probably cross-pollinated, leading to a mix of seeds on the same plant. If you plant them next year, you will probably have multiple flower colors again.

The other possibility is that the white seeds are not quite mature yet. The seeds start out white and then turn darker inside the pods when they mature. The immature seeds will probably not germinate well.

"Blue Daze" Evolvulus, a morning glory variety, can be grown in containers or in the ground outdoors. Choose a well-drained potting soil and a sunny location. If the plant has grown too large for its container, move it to a larger pot or divide it. Here's how to grow this plant in containers:

https://www.gardeningknowhow.com/ornamental/vines/morning-glory/morning-glory-in-pots.htm

Here's how to prevent plant stress when repotting:

https://www.gardeningknowhow.com/houseplants/hpgen/repotting-stress-treatment.htm

This article will help you identify the problem:

https://www.gardeningknowhow.com/plant-problems/environmental/plant-not-blooming.htm

The prolific growth combined with no flowers suggests that the plants are getting a lot of nitrogen, even though you haven't fertilized. Maybe there is leftover nitrogen from the year before, or nitrogen that migrated over from nearby fertilized plants? High temperatures can also kill flower buds so this is another possibility; in that case the plants may have delayed blooms in fall.

It sounds like you probably have high nitrogen in the soil of your new garden, which may have been added by the previous resident to fertilize their garden. Excessive nitrogen can promote strong stem and foliage growth, but will inhibit flower production, as you describe. This article explains a few other potential causes, but high nitrogen is the most likely:

https://www.gardeningknowhow.com/plant-problems/environmental/plant-not-blooming.htm

To fix this, you should first get a soil test:

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

Then, if the nitrogen levels still look high, you could try mixing in regular garden soil to dilute the nitrogen, or planting foliage plants next year to use up some of the nitrogen.

Yes, these are probably the seeds (inside small pods) and they can be harvested as described here:
https://www.gardeningknowhow.com/ornamental/vines/morning-glory/collecting-morning-glory-seeds.htm