If someone is infected a month ago with hiv, would it show up on the test and symptoms start showing?

It depends. Acute HIV infection may present within weeks and feel like a common viral infection and usuallly resolves within a few days. The disease can be diagnosed within 2 weeks by a quantitative pcr test, which measures the viral load directly. Other symptoms may occur at various times. If you suspect you have been infected, see a physician who treats HIV asap or go to a sti clinic.
Probably. Hiv antibodies can turn positive anytime from 2 weeks to 2 months. Early HIV infection may not show symptoms.