Greetings Marelise, welcome to the Forums.
The plant pictured in the OP is Prostrate Spurge (Euphorbia prostrata, formerly Chamaesyce prostrata).
It appears to be native to much of the New World, but is now also a cosmoplitan weed.
It is usually of little consequence to the growth of other plants, unless it completely covers them. Do pull out what you can, to keep its numbers in check, but don't exhaust yourself trying to futilely eliminate it.
The plant pictured in the third post is Common Purslane (Portulaca oleracea), in the Purslane Family (Portulacaceae).
it is likely native to southern Europe, Africa, Arabia, and southwestern Asia, but is now also a cosmopolitan weed, as well as being grown or gathered world wide as a useful leaf vegetable.
The plant pictured in the sixth post does appear to be an inflorescence of Alternathera, or a similar genus, in the Amaranth Family (Amaranthaceae). There are many species of Alternanthera and more character traits are needed for further identification.
The plant pictured in the 10th post is Ornamental Purslane (Portulaca umbraticola, or a hybrid), in the Purslane Family (Portulacaceae).
Portulaca umbraticula is native to the New World, from the southern United States down to South America. Garden forms have been specially selected and bred for larger flowers.