Birth rate (technically, births/population rate) is the total number of live births per 1,000 of a population in a year. It is abysmal in Japan because the population is very old, so not many people are in age of making children.
Total fertility rate represents the number of children that would be born to a woman if she were to live to the end of her childbearing years and bear children in accordance with current age-specific fertility rates. This rate is going up as Japan is slowly making it easier for couples to have children: more day-care facilities, and most importantly Japanese society is (slowly) becoming more tolerant to mothers going back to work.
Yup, but that won't prevent Japan from going right into the demography wall within a decade or two (with serious consequences on its economy, since current workers have to pay for retired ones, in a growing deficit and a climate of increasing taxes).
Yeah, and it's not a pretty way to go with solving the debt problem. This will result in higher prices for everyone, destroying savings and hurting in result the private sector. "Just" to save the government.