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You will, first, need to kill the Oxalis. If this isn't done, then you will only be spreading the plant. Disturbing the roots will cause all of the root nodules and growth points to grow twice as vigorous than before.

Killing Oxalis with chemical means will be difficult and will leave your soil toxic. Killing with boiling water will be the best option, but will need to be done several times.

Pour boiling water on the area several times a day for a few days. Once this is done you can dig up the soil. Turn it well, then pour more boiling water to kill off the remaining root nodules.

Cover with plastic for a few weeks to solarize any seeds that survive the boiling (which there will be many), then remove and plant fresh.

Here an article that will offer more information:

https://www.gardeningknowhow.com/plant-problems/weeds/managing-oxalis-weeds-in-lawn.htm

This link will take you to several articles about growing shamrocks:

https://www.gardeningknowhow.com/houseplants/shamrock

Keeping weeds out of a groundcover is indeed tricky. And oxalis has all sorts of ways to reproduce so it is almost impossible to keep up with hand pulling.

If the oxalis is in patches and not completely intertwined with the thyme, you can use a herbicide, carefully, on the oxalis. Use the foam option so as not to hit the thyme. When the oxalis is clear, you can keep it from coming back by using a garden weed preventer that is recommended to use around herbs.

Hope that helps. Here is more info about oxalis:

https://www.gardeningknowhow.com/plant-problems/weeds/managing-oxalis-weeds-in-lawn.htm

Uh oh. That would be spider mites! Here is an article that will help:

https://www.gardeningknowhow.com/plant-problems/pests/insects/spider-mite-treatment.htm

Unfortunately, your photo did not come through. Since their seeds are extraordinarily tiny, I assume you mean tubers. This is the most reliable way to reproduce them, instead of by seed.

These will need to be planted in Autumn, as the temperatures start to cool. They do not fare well in the heat. Once they sprout up for the Winter, they will be relatively low maintenance.

This article will help you with their care:

https://www.gardeningknowhow.com/houseplants/palm-leaf-oxalis/palm-leaf-oxalis-plants.htm

Here's an article that I hope will help you with your caterpillar problem:
https://www.gardeningknowhow.com/plant-problems/pests/insects/prevent-caterpillars.htm