Hi Ayesha thanks for your question.
Your 10 weeks pregnant and have some brown to white vaginal discharge
This is normal during pregnancy
Your obstetric ultra sound
is normal ,that is a good news.
Your having cough, and Doctor has advised you bed rest
, as one has to avoid medicine during pregnancy especially during first trimester, If the your cough doesn’t settles with bed ,consult your doctor who will prescribe medicine for cough. There Are medicine which are safe even during the first trimester.
Hope I answered your question