I'm not entirely certain of the answer, but I'll give you all the information I'm aware of that relates to it. There are a couple of visual sources and some written ones. The best visual source, in terms of detail anyway, is the Flemish artist Lucas d'Heere (who had spent much of the 1560s and 70s as a refugee in England). In about 1575 he illustrated a book of European costumes that included ‘Irish as they stand accoutred being at the service of the late King Henry’, an illustration of an Irish man and woman, a woman and girl, and a noblewoman, a burger's wife and two 'wilde Irische' (seems to be from a different book). There is only a single female figure in the 1581 propaganda piece 'the Image of Irelande' (in the second plate), but she seems relatively similar to the second d'Heer image, with her big sleeves and headgear. John Speed's 1621 map of Ireland has an inset with the costume of the gentlewoman, the 'Civill Irish woman' and the 'Wilde Irish woman'.

There are also some written descriptions. Luke Gernon's 'discourse of Ireland' is from 1620, some time after Gráinne's death, but still fairly relevant, as he says that only old women dress like this. The extract is quite long, so just click the link and ctrl+f 'theyr apparel' to get to women's clothes. He makes the rather important observation that 'even among theyr Irish habitts they have sundry fashions', so the pictures of generic 'Irish' women that I've posted may have little to do with what they dressed like in the West. However the description he gives of headwear from Connacht 'they weare rolles in forme of a cheese' sounds a lot like some of the pictures above (oddly enough, when looking for the pictures, I came across this picture of a fashionable Flemish lady about 80 years before Granuaile was born, wearing something similar). I find the tailoring terms a little hard to interpret, but the dress sounds pretty similar too, with narrow hanging sleeves and a low bodice, the top laced and covered with a scarf. The (pleated?) skirt isn't really recognisable from the pictures. The large crucifix also shows up in one of the d'Heere's pictures. The only mention of colour is that they prefer coloured stockings to white, and that their mantles are 'brown blew' (which I can't picture at all).

With that in mind, I can start to look at the rest of your questions (my answer mostly being 'I don't know'). While out 'pirating' Gráinne Mhaol may have worn a more practical version of clothing than most women of her class, but what changes (if any) this involved are pretty much down to guesswork, as I'm not aware of any sources that deal with something as detailed as that. For instance, she probably covered her head, as it's pretty practical (and in the very unlikely event that she was actually bald it would be warm) but would she have covered it in the sea-spray with a massive roll of linen? Probably not, but there's no clear evidence to say so. A shorter skirt, like in the 'image of Irelande' is also a possibility. Probably the only item of clothing that we can be reasonably certain about is the mantle (brat), which shows up in all the sources, for both men and women, and would be pretty warm.

Would she have worn English dress at her first marriage or when meeting Queen Elizabeth? I'm almost certain she didn't at her first marriage: even among the aristocracy English dress was very rarely worn before about the time of the plantations, but I'm less sure about her meeting with Queen Elizabeth. She certainly could have worn English clothing, and could even have had it made over there. The legendary version very much didn't (see this 18th century engraving for example), but I don't think that has much bearing on what the historical Gráinne did.

Sources: A lot of the things I linked are primary sources, so I won't relink them, but I found where they were mostly from book citations, for instance from 'Granuaile Ireland's Pirate Queen' by Anne Chambers.
The rarity of English dress was from Jonathan Bardon's 'The Plantation of Ulster' and this 'discourse on the mere Irish' from 1607/8.

edit: changed the d'Heere links to a much higher quality version.

edit2: turns out an actual 16th century dress was found in a bog. It looks pretty much as described above. Also I didn't notice it first time I went through it, but Gainsford's 1618 'description of Ireland' has a different geographic arrangement for women's headgear. According to him in Connacht they wear the ones 'like Bishop's mitres, a very stately attire' (perhaps the weird onion-like thing from the first engraving?) while the ones like 'a thick Cheshire Cheese' are worn in Munster. Make of that what you will.