Should I install a valve before this pump (picture)?

The weir on the skimmer inlet is there for a specific purpose which is to create better skimmer action. It also keeps debris in the skimmer when the pump is off. It is your option to remove it.

Suggest you continue to clean the seating area for the pump cover and lube the oring every month (as a minimum).
Also since you wish to run the pump less than 24/7, just bleed any air from the filter.
Thanks, Herman,
I'll take your advice about the frequent o-ring lubing/cleaning. I will try to do that monthly or more often and of course clean the basket at that time too.

I have the weir installed at the moment with plenty of water in the pool. With this weir, if the pool gets even slightly below half full at the poolside skimmer opening, then during the high speed of priming a vortex appears in the skimmer with air getting sucked in and causing loss of priming at the pump. If I keep the weir, I will have to keep a close eye on water levels. The vortex doesn't form without the weir unless the water is VERY low.

My other (related) concern now is that I may have a leak in the pool, despite the fact that it was replastered only 5 months ago. If it turns out there is a leak, I will probably remove the weir so I don't hurt the pump with accidental continuous air introduction when I'm not watching. Today I have started following the level of the pool over 24 hours. We'll see how much it drops.
 
With this weir, if the pool gets even slightly below half full at the poolside skimmer opening, then during the high speed of priming a vortex appears in the skimmer with air getting sucked in and causing loss of priming at the pump. If I keep the weir, I will have to keep a close eye on water levels. The vortex doesn't form without the weir unless the water is VERY low.
If you have removed that diverter in the bottom of the skimmer, that should prevent the vortex because the main drain will feed the pump.
My other (related) concern now is that I may have a leak in the pool, despite the fact that it was replastered only 5 months ago. If it turns out there is a leak, I will probably remove the weir so I don't hurt the pump with accidental continuous air introduction when I'm not watching. Today I have started following the level of the pool over 24 hours. We'll see how much it drops.
Is this new? Do you know how to do the bucket test?
 
It could be useful to do a bucket test with and without the pump running.

If you detect a leak only when the pump is running or the leak is larger with the pump running, then the leak is likely on the return side of the pump in an unseen part of the plumbing.

If you detect a leak only when the pump is not running, then the leak is likely on the suction side of the pump and would explain the air leak on the suction side as well.

If you detect the same amount of leakage with both the pump running and not running, the leak is likely somewhere other than the plumbing such as the pool shell or plaster/PVC interface.