TB is very infectious and has a high mortality rate when left untreated.
The infectivity really depends on the antibiotic resistance of the organism (Mycobacterium tuberculosis). Some strains are very sensitive (easily killed - antibiotic working) and some are quite resistant (still growing even with antibiotics). I suggest since she's still in the inital course of antibiotics, it is important to be extra cautious and treat it as infectious especially with babies/children where their immune system is weaker (still developing). There's no harm in doing this it is only for everybody's safety and im sure she would understand.

I used to work in a government hospital laboratory (TB referral Lab) in the Phils and they give free medication and all tests were free where i worked.
Make sure she takes the meds in a regular basis and not to miss it as this may result to an ineffective treatment and the bacteria being more resistant thus becoming it even more difficult to treat.

I'm not sure how often doctors request the tests during the course of treatment but when having her tests, it is advisable to have her spit out the first morning sputum as it is more concentrated (I know it sounds gross to all not used to it) and do this in three consecutive days and not three times during the day (often mistake).
Rest is vital for patients as they get weak with the disease. Home treatment is alright as long as you take all advise given by doctor to stop disease from spreading.
After the course of treatment (usually 6 mos) if everthing is negative and even culture is negative then that means she's infection free!!!