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Your both right!

If the leaves are left to die back and remain in the garden they can provide some protection for winter, but they don't fertilize the garden.
If you cut back all the plant material in the fall, the garden work is lessoned for spring and these clippings can be composted.

Many gardeners do like to leave many plants in the garden for winter interest and birds feed on many of the seeds of spent plants.
Beneficial insects and caterpillars find homes in the plants that are left for the winter.

I do some of each, cleaning up some plants and leaving many other for the winter season.

https://www.gardeningknowhow.com/garden-how-to/info/cleaning-up-garden.htm

When I moved house 9 years ago, I planted my favourite perennials in pots, not particularly big ones, just enough to have sufficient soil. Kept them until ready to put in new border. Nine years later I still have the same ones, and have given lots of divisions of them to friends. Hope this helps

Remove as much of the roots as you can, generally the root system and root ball will pull up quite easily.

You can loosen more stubborn roots with a small shovel.

Drought tolerant plants will need regualar watering until they are established.
Watering will vary according to your soil type.
Watering in the morning hours as this will mean less evaporation.
Deeper watering will help grow good, strong roots.
Here is care information for your new plants.

https://www.gardeningknowhow.com/ornamental/shrubs/rockrose/how-to-grow-rockrose-shrubs.htm
https://www.gardeningknowhow.com/ornamental/flowers/lavatera/growing-lavatera-rose-mallow.htm

Unless it is a very hard freeze over an extended period, you should not need to cover either of these perennials.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/sedum/sedums-plants.htm

Good luck with your new garden project.
Here is a link that will help you get started.

https://www.gardeningknowhow.com/special/starting/