Is it normal for air to be leaking from this part of the HVAC

Have you lived there long enough to compare the performance of the system to previous summers? (Sometimes units are undersized, or paired with poor duct design.)

Is the air coming out of the vents cold? Is it as cold as you remember before problems started? Is the airflow at the vents good?

After he added coolant, is the system working better? If yes, the system was low on charge. If not, something else is wrong.

It is possible for the system to be low on charge, but still be able to keep up on days that are moderate (albeit at a lower efficiency than a properly charged system).

It is also possible for there to be a leak, but the HVAC guy missed it. He probably only did a quick leak check (soap bubbles). There are more involved (expensive) tests he can do. However, they may not be necessary. If adding coolant made the system work well, wait and see. If the performance degrades again in a few weeks/months, you have a leak.

If there is a leak, it is usually in the outdoor unit, which is exposed to the elements, and takes a beating.

The good news is that the unit is probably still under parts warranty (typically 5 years for a cheap unit, 10 years for a good one), so if there is something expensive wrong, the parts should be covered.

(When was the last time your air filter was replaced? If it's been more than 6mo, look at it and see if it is OK. Replace if necessary. If you aren't up for replacing your air filter at least once a year, you should consider a service contract where the techs do it for you.)

I would get an 2nd opinion report. And then based upon that report make a decision on whether to move or not. A report that conflicts with the one that the landlord contracted for should be strong grounds to get you out of any lease.