#wylla

Eddard Stark claims Wylla to be Jon's mother

"You were never the boy you were," Robert grumbled. "More's the pity. And yet there was that one time … what was her name, that common girl of yours? Becca? No, she was one of mine, gods love her, black hair and these sweet big eyes, you could drown in them. Yours was … Aleena? No. You told me once. Was it Merryl? You know the one I mean, your bastard's mother?"

"Her name was Wylla," Ned replied with cool courtesy, "and I would sooner not speak of her."

"Wylla. Yes." The king grinned. "She must have been a rare wench if she could make Lord Eddard Stark forget his honor, even for an hour. You never told me what she looked like …"

Ned's mouth tightened in anger. "Nor will I. Leave it be, Robert, for the love you say you bear me. I dishonored myself and I dishonored Catelyn, in the sight of gods and men." -AGOT, Eddard II

Edric Dayne claims Wylla to be Jon's mother and also his milk brother

Brother?" Arya did not understand. "But you're from Dorne. How could you and Jon be blood?"

"Milk brothers. Not blood. My lady mother had no milk when I was little, so Wylla had to nurse me."

Arya was lost. "Who's Wylla?"

"Jon Snow's mother. He never told you? She's served us for years and years. Since before I was born."

"Jon never knew his mother. Not even her name." Arya gave Ned a wary look. "You know her? Truly?" Is he making mock of me? "If you lie I'll punch your face."

"Wylla was my wetnurse," he repeated solemnly. "I swear it on the honor of my House." -ASOS, Arya VIII

Within the series we have two different Ned's claiming Wylla was Jon's mother and I think they were telling the truth, "Wylla" is Jon's mother, not Wylla the wetnurse sure but the real mother hidden by the author. The name is chosen as a clue to point us to Jon's mother and other Wylla's in A Song of Ice and Fire support this thought.

- Wylla Fenn was a noblewoman from House Fenn. She was the mother of Lord Brandon Stark's bastard son, Lonnel Snow.

- Wylla of Wyl was a warrior-maid of House Wyl

And we have Wylla Manderly and Rhaegar Frey in Davos III ADWD

"Wylla, every time you open your mouth you make me want to send you to the silent sisters.

 "Rhaegar. We are always pleased to hear your noble counsel."

Rhaegar Frey acknowledged the compliment with a bow. He was thirty, or nigh unto, round-shouldered and kettle-bellied, but richly dressed in a doublet of soft grey lambswool trimmed in cloth-of-silver. His cloak was cloth-of-silver too, lined with vair and clasped at the collar with a brooch in the shape of the twin towers. "Lady Wylla," he said to the girl with the green braid, "loyalty is a virtue. I hope you will be as loyal to Little Walder when you are joined in wedlock. As to the Starks, that House is extinguished only in the male line. Lord Eddard's sons are dead, but his daughters live, and the younger girl is coming north to wed brave Ramsay Bolton."

"Ramsay Snow," Wylla Manderly threw back.

"Wylla has always been a willful child," her sister said, by way of apology. "I fear that she will make a willful wife."

"Wylla." Lord Wyman smiled. "Did you see how brave she was? Even when I threatened to have her tongue out, she reminded me of the debt White Harbor owes to the Starks of Winterfell, a debt that can never be repaid. Wylla spoke from the heart, as did Lady Leona. Forgive her if you can, my lord. She is a foolish, frightened woman, and Wylis is her life. Not every man has it in him to be Prince Aemon the Dragonknight or Symeon Star-Eyes, and not every woman can be as brave as my Wylla"

Your first question is answered here, but to sum up, most likely a fever. (People die from fevers and "chills" in ASOIAF all the time, they don't have modern medicine.) Because, well, Lyanna did die from a fever. And you don't have to tell people it was specifically childbed fever when all you're bringing home is bones.

For your second question, note that only Ned and Howland survived the battle for the tower. "They had been seven against three, yet only two had lived to ride away." This doesn't mention any non-combatants who might have been in the tower -- and note that Ned's thoughts about Lyanna's death actually do mention someone else:

Promise me, she had cried, in a room that smelled of blood and roses. Promise me, Ned. The fever had taken her strength and her voice had been faint as a whisper, but when he gave her his word, the fear had gone out of his sister's eyes. Ned remembered the way she had smiled then, how tightly her fingers had clutched his as she gave up her hold on life, the rose petals spilling from her palm, dead and black. After that he remembered nothing. They had found him still holding her body, silent with grief. The little crannogman, Howland Reed, had taken her hand from his. Ned could recall none of it. —AGOT, Eddard I

“And yet there was that one time…what was her name, that common girl of yours?

“Becca? No, she was one of mine, gods love her, black hair and these sweet big eyes, you could drown in them.

“Yours was…Aleena? No. You told me once. Was it Merryl?

“You know the one I mean, your bastard’s mother?”

“Her name was Wylla,” Ned replied with cool courtesy, “and I would sooner not speak of her.”

“Wylla. Yes.” The king grinned. “She must have been a rare wench if she could make Lord Eddard Stark forget his honor, even for an hour. You never told me what she looked like…”

Robert Baratheon: "She must have been a rare wench to make Lord Eddard Stark forget his honor. You never told me what she looked like."

Analogues of house Dayne

Lady Erzsebet Dayne, former heiress to Starfall and now one of the main branches most infamous members due to circumstances. A courtesan of Lys, dubbed the Dark sister due to her looks and dark past as mentioned above.

Family:

Daughter of Lord Alaric Dayne and Lady Nissa Swann

Aunts:

Lady Arissa Dayne

Lady Ashara Dayne

Uncle:

Ser Arthur Dayne

Cousins:

From Arissa:

Lady Wylla Sand born in 268

Lady Kaliandra Sand born in 270

Ser Daeron Sand born in 274

Ser Mors Sand born in 276

Lady Daena Sand born in 277

Lady Talia Sand born in 279

Ser Marence Sand born in 282

Ser Xander Sand born in 287

Lady Alyssa Sand born in 290

From Ashara:

Lady Allyria Snow born of Ned Stark born in 282

Prince Valarr born of Prince Aenar Targaryen born in 284

Prince Matarys born of Prince Aenar Targaryen born in 286

Princess Alyssa born of Prince Aenar Targaryen born in 287

Princess Maegelle born of Prince Aenar Targaryen born in 289

Princess Rhaena born of Prince Aenar Targaryen born in 290

From Arthur:

Lady Azaria Sand born in 276, before Arthur joined the Kings guard from a relationship with a common woman

Children:

Ser Edric Dayne, heir of Starfall born in 287 and father is the eldest son of the prince of Lys

Lady Clarisse Dayne, born in 287 and father is the eldest son of the prince of Lys