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In most areas it will be too late to plant them. To store them until the spring, place them in a cool, dry location. The vegetable drawer in your fridge is a good place.

How is your soil's health? Have your fertilized? The lighter, or looser, the soil, the larger your shallots will grow. They also need to be fertilized with compost, which helps the soil as well.

The tops will regrow as long as the bulbs are still healthy. You may, however, need to do something about the rodents, if indeed they are the problem.

When to harvest shallots can be tricky for some, as this usually depends on when planting took place. Generally, fall plantings are ready to harvest in winter or spring while those planted in spring may be harvested in mid-summer to early fall.

Harvest your shallots when the bulbs are about a quarter inch around but wait for the leaves to yellow before lifting. For an extended harvest season, plant and harvest the largest shallots first, replanting smaller bulbs in their place for harvesting later.

It could possibly wireworms. If you see tunneling that is a good sign it is them.

More likely it is root rot. Shallots need excellent drainage to do well. If it was too wet in the area where they were growing, then it would cause problems like you describe.

It may have been a lack of phosphorous or it may have been heavy soil or it may have been a lack of water. All of these can cause undersized bulbs on shallots. Have your soil tested and amend for phosphorous as needed. If you have heavy soil, try amending with organic rich material like compost. And make sure that your shallots are getting enough water, deeply enough throughout the growing season.