If she is at home, serving tea, then no she would not wear a hat. If she was visiting someone for tea or having tea in public, then yes she would wear a hat. Don't forget to add a veil to her hat if she is in a car. Sorry, I love those types of hats from the turn of the century.

She would carry a purse to someone's house or out in public. She would wear gloves as well. Gloves may or may not be worn at home to serve tea and may depend upon who she is serving.
At the turn of the century, undergarments and clothing were in transition. By then, panalets/drawers were known as bloomers, so you are correct in that term. At this time she would wear those and a camisole, plus a corset, petticoats, and stockings. I'm not sure if corset covers were out by then since corsets were mass produced and readily available--protecting them was no longer a priority.
Even though this young lady is not of high standing, the lower classes took their cues from the upper classes. The main differences in dress would be that a young lady of lower standing would buy mass produced clothing, where as a lady of the upper class could still afford to have clothes made for her by a modiste (dressmaker).
I don't know where you are, but museums that house good costume collections are always fun to visit. I'm thinking specifically of the V&A in London which often has even the undergarments on display. Also the Met in NYC has a good collection as does the Smithsonian.
I have a lovely green silk evening gown from this period. It's trimmed in black lace and is shot through with gold thread. The detail always amazes me when I pull it out of storage!
Good luck!
A young lady, not of high standing, wears a yellow chiffon tea gown to serve tea to a gentleman in Boston. What else is she wearing? Does she wear a hat? Does she wear gloves? Shoes? Stockings? How high? How are they held up? Are they embroidered? The more specific the better. (She happens to have some experience in millinery, so I can go to town on the hat, if she wears one.)
What is the garment closest to her skin? If she removes her bloomers will she be exposed?