Clover can have long stems. This one looks very healthy and will out compete your thyme and so I would carefully remove it.

It's "common ivy" - Hedera helix. It IS difficult to remove because it forms roots at nodes that cling on to rocks and also root into the soil. However you can get rid of it with persistance pulling/scraping it off, or you can use a weedkiller if you don't mind chemicals. Here is a link that might be useful: weediness of hedera helix

Might be impatient - but what ever it is it is not happy. I would cut it back to 3 or 4 nodes each stalk and put the cut pieces in either water or pearlite - water works just fine. cut into lengths of 4 or so leave nodes or no less than 6 inches. Change water at least every two days and put is bright light on window sill. Very soon it will get calcified nodules and little roots will come out of the stem and you will have lots of new plants. When the water roots are about 4 - 6 inches - put into soil and keep moist and they will be ready for spring. Be patient - if you keep the water from getting slimy and toss out the few that sort of slime up you will have enough for you and your friends. LOL These are garden plants - not house plants in most zones. Good luck and keep us posted.

To be a bit more specific than Tiffany, the bottom one is a red maple, Acer rubrum. The top one looks to be a birch, and probably a gray birch, Betula populifolia, though I am not positive on that ID.