|First to the time and date issue. All computers have a small 3.3V battery inside them to maintain the settings, including the date and time. It is possible the battery is OK but you have not accessed the BIOS to reset the date & Time. Watch the screens at start up for instructions on what keystroke/s to hit in order to enter the BIOS (setup) screens.|
Next, the hot issue. Laptops are prone to overheating. The only access to fresh air is the grill type vent on the bottom. Dust and dirt can enter there and clog things up. You can buy canned compressed air to blow out the dust and dirt. If that doesn't work then I suggest you take it to a shop for dis assembly and cleaning. Also, ironically, laptops should never be used on your lap. Always place them on a hard flat surface so the air can enter the vent freely. You can also purchase a laptop cooler, to place under the laptop. You should hear the fan when running and should be able to feel the hot air being discharged. The fan only runs when necessary in order to conserve battery life. If you only use it on AC then you can change the settings.
Finally, it is probably the case, but you may want to read the manual for the laptop to verify the integrated video is actually broken. If the screen is visibly damaged then that is that. However, there may be a setting that turns off the laptop screen and it is set that way.