I jumped the truck the other day and this connection marked with the red arrow broke along the line, you can see where we jimmy rigged it and used zip ties to hold it onto the plastic piece long enough to get home. I want to know what this plastic piece (yellow arrow) is and if I can bypass it, or at least bypass the heater (upper) hoses and directly connect them.

 

That's your heater valve. Just get a new one . They're not that expensive.