Oh great, i was worried you fried the ESP-01, they are quite sensitive.
Yep GPIO 0 needs to be HIGH or free-floating at boot. (you can use it anyway you want after that)
Keep in mind that the LED_BUILTIN is set as 2 (referring to GPIO 2) as a default in the IDE, but usually an ESP-01 has it on GPIO 1 (the TX-pin)
Well of course i do ! First of all, on the ESP side you should be using hwSerial, you have only 4 GPIO's available, and since 2 of those are connected to the UART, it makes sense to use it. (less CPU intensive and more reliable)
On the UNO side you could also use hwSerial if you aren't using it for something else that would be the preferred choice. All you need to do for that, would be to swap the TX & RX on the UNO side.
May you decide that you want to use swSerial anyway, you should use a lower BAUD-rate for that, say 9600kbps
Sending data is generally fairly straightforward, it is the reception that is tricky, but have a look at Serial Input Basics By Robin2.