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As long as the ground is still manageable, you should be ok with digging them up, but as far as replanting them, I would place them in containers instead. Storing them in the garage should be fine. With exception to any bulbs or tubers, they may require a bit of water once a month throughout winter. Keep the bulbs/tubers dry, however, and store them in a cardboard box with some newspaper or peat moss in an area that will remain dark, cool, and dry. Once spring returns, you can replant in their new location and care for the same as before.

Always pot up to the next size. If you have a 1 gallon pot, go to 2.

This depends greatly on the plant type and your growing zone.

Here is a link to help you determine your growing zone.

https://www.gardeningknowhow.com/planting-zones/

If you have the specific plant type we can give you more specific information on care and when to transplant.

Make sure you check the seed packets for information on the temperatures needed for germination. Not every seed needs the same temperature.

Most germination heat mats have a reostat that can be changed to the proper temperature.
Do you have a way to control your heat wires you mention?

Since I'm not clear on the type of 'heat wire' you mention, it is difficult to make a recommendation.
When using a seed heating mat system the pots or germination trays sit direction on the mat.
https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/cheap-seed-starting-how-to-germinate-seeds-at-home.htm
Make sure you also have ventilation in your cold frame.

https://www.gardeningknowhow.com/garden-how-to/projects/building-cold-frames.htm
https://www.gardeningknowhow.com/special/greenhouses/how-to-use-cold-frames.htm

The most you can do it assure you have planted in the correct soil and growing conditions.
Transplanting during active growth and blooming is not in the trees best interest, but I understand you have to do what you have to do!
Watering a newly transplanted tree is the most crucial step. Daily water for the first 2 weeks. Soil should be moist but never soggy.
Apply 3 to 4 inches of hardwood mulch to help with moisture retention.
Do not try to fertilize into health. This would further stress a tree.

Good luck!

I always provide shade for newly transplanted plants for 3 days. This gives the roots time to recover from transplant shock. Anything that casts shade, like a cardboard box or leaf bag over a stick or shovel, can be used to create shade. Keep the plant shaded until it looks better. Water when the soil feels dry 1-2 inches down from the surface. Oddly enough, over-watering causes similar wilting to under-watering. Wind and hot weather increase water needs. The rocks in your garden hold heat and heat up the soil. If you are in a hot weather state, consider using wood mulch in place of rocks. Flowers are very draining on plant reserves. Remove the flowers on your stressed plants to make recovery easier. Another consideration is whether weed killer was used in the area this year or last. Some are persistent in the soil; glyphosate is not.