A lot better to purchase a programmer or even an esp8266 like nodemcu.
1) you have to remove it or at least keep the atmega328 in reset
2) if I remember correctly, you connect gnd-gnd, rx-rx, tx-tx
3) a short period with 5V on rx won't probably fry the esp but you can insert a 4k7 resistor (in rx-rx) to be on the safe side.
4) the 3V3 from your UNO might be sufficient for the programming, for "test-runs" you definitely need an cap on the esp-01 (min 470uF/6V) or better seperate 3V3/1A supply.