It sounds like your battery is having problems. Even if Apples Battery Health meter shows it is at 80% health, the battery could still be bad. I have an iPhone 6 that says it has 86% of its battery health left, but it cannot sit for even a day before it is dead. I would suggest a new battery, as it could solve these problems.
It is possible that this model doesn’t have a battery. Many calculators are 100% solar powered. However, if this model does have a battery, the battery is probably dead and needs to be replaced. These batteries are usually not rechargeable even though there is a solar panel. Over time, the battery will die. If there is a battery compartment or cover on the back, then it is simple to replace the batteries. Otherwise you may have to take the whole thing apart (assuming it has a battery in the first place). Hope this helps.
What model number is on the back of the display? Instead of looking for a display for that specific laptop, search the model shown on the back (or wherever it is located) on the original display that you removed. Try searching for that model number. Many displays are shared by multiple devices, it should be easier to find one.
This sound like a battery issue. I have seen this problem before. The battery is dying so quickly that it completely drains, resetting the date and time. The date and time is only kept if there is battery power. I doubt it is a motherboard issue, I would try a new battery and see if that fixes it. Hope this helps.
There are lots of phones like that. For example the Samsung Galaxy S4 had different versions as well, weather it was a Verizon locked phone or other carrier or from a different country. What the “locked” means is that the phone is locked to that carrier. This means you have to use that carrier that phone is locked to in order to service it. If the phone is Verizon locked, you can only service it with Verizon. If the phone is “unlocked” this means that you can use any carrier. It is not locked to any specific one carrier and you can service it with Verizon, Sprint, AT&T, etc. Hope this helps.