Which Component Is Broken in the Refrigerator
The easy thing to try replacing is the thermistor, they are cheap and you just cut the old one out and then crimp a new one on the wires. Biggest challenge is usually finding where they are hidden in your fridge (you'll often have 3, maybe more, and any of them can be bad), but it looks like you've opened things up enough to see them. Look for any small piece of plastic on the end of two wires, I've seen them 1" long by 1/4" in diameter. They'll be clipped on to the defrost piping, and probably somewhere else high, possibly behind the track that the shelves hang on, or near a vent.

I'd also check all the gaskets on the door to make sure you're getting a tight seal. If these first two checks come up empty, then I'd personally throw in the towel and buy something new assuming the fridge is older than 8 or so years.

If it's an old fridge, then you may be looking at low refrigerant and that isn't typically fixable in a refrigerator. The ice is an indication to me that this could be your issue, and if so, I'd find out if there's any way to recharge the refrigerant on your unit and that the repair man can do this before making a service call. Otherwise, that's just throwing away money.

One last possibility is a bad control board. They can be swapped out, but aren't cheap compared to the price of a new fridge, and I think the odds that this is your issue are relatively small. If you attempt to replace this, consider buying one from somewhere that lets you return it for a refund.