Oh, my. Those are just beautiful!
I'm not sure if the tie quilt should be washed or dry-cleaned. Do you know if your mother pre-washed the ties? If she didn't, or if you don't know, I would do more investigating before deciding how to clean it. It's because the ties are probably make of both silks and blends.
For the other quilts, machine washing and drying are the way to go. A front-loading washer is best because it is easy on a quilt. Laundromats have front-loaders, but to be safe it's a good idea to run a cycle on the front-loader with plain water first to make sure everybody else's soaps, etc., have rinsed away. Then I would use cold water and Synthrapol to wash one quilt at a time.
If you have a top-loading washing machine, be aware that the machine agitation is hard on a quilt. You can still use a top-loader, but stop at the agitation cycle and just agitate by hand, pushing the quilt down into the washing water for a few minutes to work the soap and water through. Drain the machine, add rinse water, and do the same for the rinse cycle -- skipping the agitation and just pushing the quilt under by hand. Drain and do a second rinse cycle, then spin dry. Spinning is not hard on a quilt; it's just agitation from a top-loading machine that is hard on it.
Whatever machine you use, you need to watch the quilt and remove it as soon as the wash cycle is done so wet fabric is not laying against wet fabric (which can result in color bleeding). I recommend Synthrapol because, if there is any unset dye in one of the fabrics, Synthrapol will keep the unset dye suspended in water so color does not settle in another fabric. Synthrapol is widely available in quilt shops or online and is not expensive.
What kind of dryer do you have? Mine is a very small front-loader and I would not want to stuff a large quilt into it. You may be better off with a laundromat dryer. You do not want to dry completely. It's best to take the quilt out while still somewhat damp and then "block" it to finish drying. To block, you lay the quilt on top of a sheet on top of carpet, square the sides and corners and pin them in place, then place a fan nearby to finish drying. If the quilt is completely dry, you can spray it with water to dampen it enough to be blocked. Blocking makes sure that the quilt ends up perfectly square, the way it started. You can also skip the dryer and just block the quilt and let it dry under the fan; however, this can take a long time and the quilt needs to be turned over.
Never expose a quilt to sunlight if you can help it. Sunlight can fade a quilt very quickly. Also never hang a wet quilt over clotheslines; the weight of the wet fabric can break stitching lines. If you ever do want to dry a quilt outside, lay it out on top of a large sheet and then cover it with a couple of sheets to keep sunlight off it (not to mention bird mistakes).
Good luck!