Esplora is based on a MCU with native USB. The native USB reports connected only when it really connects. You have in setup() a while loop until the USB Serial is valid. The skech waits there for connection. Comment out the while (Serial) loop.
The TX and RX LEDs of Esplora are connected to the native USB port Serial interface. They will not light for the ...
You could simulate the switch with the button by using a state variable that hold the state of the "switch". I don't know your code, so I can only give you an example on how to do that:
// the includes are here, if you need any
int flagActive = -1;
// ... set the internal LED to blue
if ( < switch pressed > )
With those technologies you have no chance. They are both low-bandwidth devices suitable for sending small amounts of sensor and control data. They can in no way be used to send video data. Both devices are rated in the kilobits or (at most) hundreds of kilobits. For video you need tens or even hundreds of megabits (depending on resolution and compression)....
It will not work. USB communication is master/slave. There must be a 'host' and a 'device'. Uno and Esplora are two 'devices'.
Connect the boards over TTL Serial. Wire Rx and TX pins together, TX to RX and RX to TX. And ground of course.
Serial on Esplora is the USB interface. In 'core' I see the Esplora has a second serial interface called Serial1. ...