It rejects "I am wearing a jacket" even though to have a jacket on implies one is wearing it.
Different languages express the same idea in different ways. Here Irish expresses it as 'There is a jacket on me' whereas English would express it as 'I am wearing a jacket', 'I have a jacket on me', 'I have a jacket on', 'I have on a jacket'.
In Ó Dónaill's Irish-English dictionary under meaning II(e) for the preposition ar there is the following:
(e) (Of anything covering or enclosing, attached to or supported by body or thing) Tá clár, clúdach, air, it has a lid, a cover. Tá úlla ar na crainn, the trees are bearing apples. Tá mála mine air, he is carrying a bag of meal. Bhí culaith ghorm orm, I was wearing a blue suit. Dá mbeadh trucail ar an asal, if the donkey were drawing a cart.
Yes. The tenses are different in Irish, as they are in English. Caithim seaicéad = I wear a jacket (habitually, generally); Tá mé ag caitheamh seaicéid = I am wearing a jacket (today, right now). Note that seaicéad takes the genitive form seaicéid after the verbal noun ag caitheamh.