Its their standard customer doc, our service desk agents would send them out to customers but its variable how much use they were, sometimes the end user would have not plugged the charger in correctly or similar, the call would often land on the desk of my teams.

With chargers if you know someone with a multi meter you can check if its got an output, we'd measure them in my workshop team but we had so many in our stock that if we thought the charger was failing for any reason we would throw them away and replace them, if no one has a meter or there's not a PC shop nearby who could check it, perhaps buy one anyway, see if it charges and decide what to do with your battery at that point.

I'd say it's perhaps a case of weighing up the time/cost and value of the laptop, the #1 issue we always had with laptops was failed chargers, so much so that if we suspected one we would just ship one to the customer, if a laptop came into the workshops and it had a 3rd party (non HP) charger, it would go straight in the bin and we would supply a HP one, then we'd test.

As for the charging port, it's a tricky one, there is often a little bit of movement in the connector but it shouldn't wobble too much, if it is damaged they can often be replaced, it's another case of time/cost versus the value of the laptop.