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.
 
Is this new? Do you know how to do the bucket test?

I feel like it is relatively new. I have been noticing the need to add water more often since the pool was replastered about 5 months ago. I have been running the pump 24/7 since replastering. However, I am much more attentive to the pool since replastering, so that could play a role in what I observe.

I am aware of the bucket test since I have recently started researching how to detect and find leaks, but the air in the pump basket has been taking all my attention lately.

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.
Thanks mas985 for the concise summary.

I should have started a bucket test yesterday, but I first just did the simplest thing and marked the pool side water level with tape. It has been about 20 hours since I marked it, and the level has dropped 1/4" in cool weather (mid-50s low, mid-60s high) with the pump OFF.

I did a short, 2-hour test yesterday with the pump ON, and it lost about 1/8", so over 24 hours that potentially could be 1.5 inches! I will need to confirm this with the bucket test. It's sounding like a pressure side leak, but we'll see.

I will start the bucket test later today, with the pump ON, when my first 24-hr (non-bucket) test finishes. I'll need to repeat it with the pump OFF. I'll report back what happens.