As-salāmu ʿalaykum wa raḥmatullāhi wa barakātuh, After wuḍū’, I wore socks and later wiped over them in a new wudu. If I then removed them while still in wuḍū’ and put them back on without breaking it, can I still wipe over them? And does the period start from the first wipe? Jazākallāhu khayran.

The period starts from the first time you wiped over them. But, since you took them off, your wudu is not invalidated according to the correct saying of the people of ilm. However, the wiping is no longer valid. What does this mean? It means you cannot wipe over them again. You need to make wudu and then wear them upon purity; then you can wipe again if you want to. Otherwise, no.