Adding sink next to laundry drain

Here is my current setup i have behind the wall



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!

The word is "advice"
No time this morning to do long answers, but short answer: don't use sharkbites on soft copper tubing. It does not hold shape properly.
Adding a sink to that setup: forget about it without coming up from the floor and redoing the whole set up.
Hire a plumber to have it done properly, and get a permit.

How can I connect PEX to the soft copper tubbing?

Solder on a pex adapter

Any reliable way of connecting it without soldering?

What's underneath the floor. Basement, crawl space?????

Normally I would suggest you get a torch, solder, flux, sandpaper, and learn to solder. However, I have had lots of trouble soldering old corroded copper like that. You really have work hard to get down to clean shiny metal and it can be a pain if you haven't done it before.
Go with brass compression fittings instead.