The analyses of a text #7

The narrator recalls his adolescence.
His young wife died from died of illness, and His old friend Steerforth just now drowned.
He came to his house to break the news to his mother.
Rosa Dartle is Steerforth's cousin.

Chapter 56 THE NEW WOUND, AND THE OLD

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At her chair, as usual, was Rosa Dartle. From the first moment of her dark eyes resting on me, I saw she knew I was the bearer of evil tidings. The scar sprung into view that instant. She withdrew herself a step behind the chair, to keep her own face out of Mrs. Steerforth's observation; and scrutinized me with a piercing gaze that never faltered, never shrunk.
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'My son is ill.'
'Very ill.'
'You have seen him?'
'I have.'
'Are you reconciled?'
I could not say Yes, I could not say No. She slightly turned her head towards the spot where Rosa Dartle had been standing at her elbow, and in that moment I said, by the motion of my lips, to Rosa, 'Dead!'
That Mrs. Steerforth might not be induced to look behind her, and read, plainly written, what she was not yet prepared to know, I met her look quickly; but I had seen Rosa Dartle throw her hands up in the air with vehemence of despair and horror, and then clasp them on her face.
The handsome lady - so like, oh so like! - regarded me with a fixed look, and put her hand to her forehead. I besought her to be calm, and prepare herself to bear what I had to tell; but I should rather have entreated her to weep, for she sat like a stone figure.
[David Copperfield by Charles Dickens]
1. I'd like to know if "that" means "in order that."
2. I'd like to know if "plainly written" modifies "what."
3. I'd like to know if "what" indicates Rosa Dartle's look.
4. And I'd like to know what "so like" means.
Thank you in advance for your help.