Aquarius this month

The Sun enters sidereal Kumbha at Kumbha Sankranti, around Feb 13, and leaves about 30½ days later. Kumbha Sankranti is the sankranti after which the Kumbha Mela's bathing dates are reckoned when Jupiter and the Sun align appropriately. That month is the Sun in the 1st house from your rashi — traditionally a period of exposure and visible effort rather than ease, since the Sun in the 1st tends to burn as much as it illuminates.

Because the Sun is the graha whose sign changes on a monthly cadence. The Sun crosses one rashi in about 30½ days — sankranti to sankranti — so a solar month and a change of sign are the same event. The Moon changes sign every two and a quarter days, which makes it the driver of a daily or weekly reading, and Jupiter takes about one year per sign, which makes it the driver of a yearly one. Each timeframe belongs to whichever graha moves at that speed; that is what makes the three readings different in kind rather than just in length.

No, and the difference is why the Hindu calendar needs an extra month periodically. A solar month runs from one sankranti to the next, about 30½ days. A lunar month is the synodic period, new moon to new moon, and that is 29.53 days — split into two pakshas of fifteen tithis each, the bright Shukla and the dark Krishna. A tithi is a thirtieth of that, about 0.98 days, which is why tithis drift against calendar dates. And the sidereal month, the Moon's actual circuit of the zodiac, is shorter again at 27.32 days. Twelve lunar months fall about eleven days short of a solar year, so an intercalary Adhika Maasa is inserted roughly every 32 or 33 months to keep the two in step.

The classical answer counts from your rashi rather than naming a calendar month. The Sun in the 11th house from Kumbha — that is, when the Sun is in Sagittarius — is the most favoured of the twelve for gains and income, and the 3rd and 10th are also good, because the Sun is a malefic that strengthens in the Upachaya houses. The months to be careful in are the Sun in the 4th house (Taurus), which unsettles domestic matters, the 7th (Leo), which brings friction in partnership, and the 8th (Virgo). The full table on this page gives all twelve.

Your Moon sign. Vedic astrology reads the Chandra rashi as primary — the reverse of the Western convention — and gochara, transit reading, is counted from the Moon sign specifically. Everything on this page numbered as a house from Kumbha assumes Kumbha is your Moon sign. If you only know the sign matching your birth date, that is a tropical Sun sign, and on the sidereal zodiac it is frequently a different sign again. A free Kundli gives you the Moon sign and nakshatra to read from.