Hi.
1- this doesn't matter too much really. You can almost say either but 'my phone screen is cracked' sounds most natural to me, although 'my phone's screen is cracked' is technically correct. there are some instances of this like 'car door' where not adding the possession (car's door') is especially common. Also, for instance, 'computer screen'. I would generally side with 'car door', 'phone screen', etc.
2- some of your examples are more correct than others. 'my dog got sand all over my car' is perfect.
'my ceiling got a leak in it' is also correct if you wish to talk about the actual event of the ceiling 'springing a leak'. (if you don't know the expression- 'the pipe has sprung a leak' or 'I cracked the bath tub and it sprung a leak' are other examples). But if you wish to talk about your ceiling NOW, you would say 'my ceiling has got a leak'.
3- Either of your examples here make sense. If you say 'I got water all over my room' it implies that you directly caused it by, perhaps, spilling a drink. 'my room got water all over it' doesn't imply that you caused the room to become wet.
Also, you say- 'you FORGOT to close the window', 'when you GO into your room' and 'the water keeps coming in' doesn't imply so much that you are trying to stop it coming in. If you are trying to stop the water you say 'it keeps GETTING in'.
English is tricky (AND VERY SILLY AT TIMES!) but none of your mistakes here are serious at all.