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You can, definitely, save this and keep it going indoors. The first thing that I would do is repot it, though.

This article will give you information on caring for these in container: https://www.gardeningknowhow.com/edible/herbs/lavender/growing-lavender-in-pots.htm

Place your potted Lavender in good location. This means as much sunlight as possible with good air circulation. A sunroom is great for this although you could rig some broad spectrum or fluorescent lights up in a basement if you cannot find a sunny spot. Houses in winter tend to be drier due to heating so humidity usually is not a problem.
Make sure to check on your Lavender plant periodically and as for watering try to be lax about it erring on less and not more. Allow the top inch or so of soil to get bone dry. Rotate the pot every week so it receives uniform light and air.

Over watering potted Lavender grown inside is the main reason for plant fatality. Your plant will not need as much water and definitely not any fertilizer until warm spring weather returns. Be sure to wait until the last spring frost date for your area or you overwintered your potted Lavender for nothing!

https://www.gardeningknowhow.com/edible/herbs/lavender/growing-lavender-in-pots.htm

Although they are far less likely to germinate, you will still see a very high germination rate, although it will be a slow process. This article will help: https://www.gardeningknowhow.com/edible/herbs/lavender/lavender-seed-propagation.htm

Did you spray because you suspected spider mites, or that the plant already had them? If there were already spider mites, then this could be a problem. As they suck sap, they also introduce their own poison, which if left treated for too long will destroy a plant beyond repair. If you water with a few drops of rosemary oil in the water, then this will deter spidermites. You should only have to do this once a month or less. I would say just keep the soil slightly moist with a short dry period in between. This might keep it alive until you can put it back outside.

If you are using simple Dolomitic lime, then most plants, unless they are specifically acid loving (and even then, sparing amounts won't hurt) can benefit from it. Hydrated lime can be a bit rough on some plants, so you will need to be much more careful with this.

Here is a list of plants that highly benefit from the alkaline conditions that lime has: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/plants-for-alkaline-soil.htm

This can, actually, mean either! You can prune 1/3 of the length of the entire plant, or you can choose to take out 1/3 of the plant by limbs. This will be your preference.