Adding sink next to laundry drain

[J Fiddy]

Here is my current setup i have behind the wall

https://ibb.co/naHO4z

Is the copper connected to the hose bibs a specific kind that is bendy like that? It has a few leaks and need to replace some of it. Can I use push-to-connect fittings on that copper, and what are your thoughts on using that technology? Also, how can I safely run the drain to the right of the existing washer drain and properly vent it?

Thanks in advance for your advise!