Was it an original power cable? Generic brands can be very glitchy to keep a monitor powered on. One thing you can check is the voltage if you have a multimeter laying around. Otherwise the bad power supply could have shorten out the monitor as well.
The difficult part can be to determine which part is actually failing because of the water. It could be something to do with the graphics from the main logic board but is likely to be just the LCD if it is just a few lines. If it was pretty recent still there could be moisture still inside that is creating the glitch as well. So might be able to try putting the phone in the rice for a couple more days and restart it after to see if it helped any more.
There is a program called Macrium Reflect. There is a free version. I am unsure if the force clone is an option with the free. If so you can clone all the data that's on the failing hard drive over to the new hard drive. Some of it could be corrupted but from my experience in cloning hard drives it does the trick fine. Otherwise it might be worth looking into other programs that can clone a hard drive. You would want to search for one that can clone bad sectors. Hopefully it works for you!
The beeps are known as the beep codes. Its the computers way of telling you what is wrong with it. Make note of how many normal beeps it does or long beeps. Look those up according to your computer model and see what it says. Can remove the battery and ram. Hold power button for 15 seconds. Then plug in the charger with 1 stick of RAM in. See if it does anything from there. If you removed the processor then go ahead and remove it again and make sure its clean. Make sure to replace the thermal paste with new thermal paste. Otherwise you risk overheating the processor.
Try to remove the battery and hold power button for 10-15 seconds. Put back in the battery and try turning it on again. If that still doesn’t work verify the charger is charging and matches volts that the laptop requires. If it still doesn’t work or show any signs of activity it would most likely be shot. If you wanted to repair look at pricing for motherboards.
Touchscreen or the LCD? With this model you should be able to pry around the displays plastic bezel and the back LCD cover. this will allow you to get the front screen off. If you need the touch then find another one of those. There should be model numbers somewhere on it to find. if the LCD then remove the supporting screws and then pull it up slowly. Disconnect the cable from the bottom of the LCD and then find the LCD. Should have part numbers/model numbers to look it up.
Sorry to hear you had a bad IT experience. The wires for the antenna you can usually find for that laptop pretty cheap. The front digitizer can be popped off. You can do so by taking a thin flathead screw driver and prying around the edges of the plastic bezel and the back LCD cover. Take a look at the antenna part and see if there is a part number or model to do a search and find one of the exact same model.
This should have a detachable keyboard correct? If so look up a new keyboard for the surface. Make sure it is exactly the same. If not then try prying a little bit more to see under the trackpad. Be careful not to break it anymore. Is anything stuck under it?