I have a couple of questions regarding smartphones

This question sounds a lot like some kind of homework. Why are you asking these questions?

Someone told me that you need to find which pins are + and - on different components for exemple battery connector or charging ports. Why is that?

This question can’t be answered in a meaningful way without more context. I’ll ask you a question “What do you think the difference is between positive and negative wires in a phone?”

What does it means when the multimeter beeps on continunity when checking one pin of the battery connector and yet the rest dosent. Checking them all against ground

It means that pin is continuous with ground, aka that pin IS ground.

How do you close a circuit with a multimeter on a smartphone board and why do you need to do this

Closing a circuit just means putting a wire to connect two spots, same thing that happens when you flick a lightswitch, or lower a drawbridge. You ask how would you do this with a multimeter—-answer: you don’t.

When you remove a bad cap. Resistor or inductor which causes shorts. Do you replace it with a new one? If so. Where do you find new caps resistors and inductors that fit the specific phone your working on?

Resistors and inductors can NOT cause shorts, so removing them to solve a short would be bad repair. Capacitors do fail in a way that they become the reason a line is short, so removing a bad cap to clear a short is reasonable. Replacing it depends on the function of the cap, and the difficulty in sourcing a new one and the technical challenge of replacing it. That is—if it is easy enough to replace then do it. If it will risk the whole device because clearance is limited, then skip it.

You find new ones at the cap store—mouser.com