Why does my baby sleep for less time?

Hi doctor,

My baby daughter is 3.5 months old and weighs 6 kg. Her sleep lasts only for a few minutes. Overall on average, she sleeps 8 hours a day. She cries a lot, always remains fussy and irritated. She takes formula milk as well as breastfeeds after every one and a half hours. Her demand for milk is very high. When my wife does not meet the needs, she takes formula milk. We make her burp every time she feeds to ensure that she does not have any irritations. A few days ago, the doctor gave her Barbit 30 (twice daily) for seven days. She was alright at that time, slept peacefully, and was not fussy. But now, she has returned to her previous form. Her physician has asked to continue for another five days and stop. Are there any sleep techniques that can be executed? Her mother and I have gone restless. She will get her vaccines this week. Can we give her Barbit 30 besides tablet Napa after the vaccination? We are worried about her health if she overweighs due to the excess feed. We would also like to trim her hair. Is it safe to use a trimmer for her at this age? I am concerned about her scalp if it is sensitive?