Plumbing – Copper tubing on refrigerator is leaking

Yes you can do that (it's the standard way of converting to/from copper), and it shouldn't cause any troubles (read Why would you use copper over CPVC or PEX? and Does Pex tubing have any advantages over copper? for some comparisons).

Of note: you can get the male adapters that fit on the outside of copper pipe, and also that fit on the inside of copper fittings (like a street elbow). Can be handy sometimes if you're working in a tight space, or were going to immediately make a turn anyways (since you're soldering the fitting anyways, it's barely any extra work to do an elbow+adapter, and saves a PEX elbow and couple of crimp rings).

You branch off from the copper tubing with a 1/4" compression tee. Cut out a small section of tubing for which the tee replaces. You must use a tubing cutter to do this, any other method will cause the tube to become oval.

Install the short piece removed on the side outlet of the tee. Attach a 1/4" compression x 1/2" pipe thread adapter to the short section. If you can find a tee with a pipe thread side outlet (preferred), skip the previous step.

Connect a 1/2" pipe thread to hose adapter. When buying parts, ensure the male and female parts mate correctly. If you can't find the proper mates, get a short nipple or coupler to change gender. Use several layers of teflon tape on male pipe threads, wrapped in the direction that causes it to get more snug when the pipe is screwed in.

Attach hose and controller. After all this, there probably isn't much pressure left, meaning you could probably transition from hose to 1/4" drip fittings for the run from controller to plants, keeping the installation less obtrusive. If you use pressure adaptive drip fittings, you can better regulate the flow to each plant, otherwise the first plant will get the most water, the last hardly any.

Be careful, though unlikely in this case, as high pressure can cause drip fittings to blow apart. You also can't use most pressure reducers, as they can spill water as they function. You'll probably have to go from hose to 1/2" drip main to 1/4" drip distrubution. The 1/2 to 1/4 punched connection can leak slightly, so it should be done over the first plant.