The heat pump can move heat indoors to outdoors ("air condition") as well as from outdoors to indoors (heating). The cooling only unit can only air condition.
If the house already has heating or you rarely need heat, you can save some money by getting a cooling-only unit, and maybe supplement that with portable electric heaters as needed.
As to brand comparisons, I strongly recommend you review the Consumer Reports reader reliability report of air conditioners and heatpumps: that can be done online by paying a fee, or visiting a library.
As for some installers carefully measuring and the others not, the latter could be fine if your dwelling is not especially unique: an experienced installer has run the numbers so many times that they know what the result is. That experience is helped by there being a coarse choice of capacities: 2 ton, 2.5, 3, 3.5, 4, 5, 6, 7.5, 10, 12.5, etc.
On the other hand, not measuring could be a sign of an unknowledgeable installer. Do they also propose a 3.5 ton unit?
As far as questions to ask, I like to elicit being assured that if anything goes wrong for X years, they will fix it for free. If they can say so confidently, that is a very good sign that they don't think anything will go wrong.
It might be useful to ask an installer why they choose 3.5 tons and not 3 or 4. There is a certain degree of assumption about worst case and average case conditions. If the a/c can't quite keep up on the 5 hottest days that is one thing, but if it can't keep up on 30 of them, that is quite a different matter. Likewise, an oversized system can chill the air so much that it doesn't have to run much, making the air stagnant and possibly muggy. Questions asking about how they balance those factors are good for you to understand and for the installer to reflect upon.
You've got a wiring problem, or a bad circuit board somewhere.
It's possible that the condenser unit is wired to run all of the time, but some protection circuit is kicking in and shutting it down. If that's the case, you would probably have ice build-up on your coils.
What is more likely is a relay somewhere is sticking, and remains closed when de-energized. If you provide the model number of you air handler and your condenser, I might be able to provide more direction.
Best Answer
With a 1500 sq ft home a 2.5 ton should get it nice and cool. Things that may affect your ability to cool, ducting not insulated, home not insulated, extreme heat (I don’t consider 90 extreme)
You did not state what the unit did cool to. Without this we cannot calculate the delta to tell if a larger unit would work for your home. The tonnage of a unit is how much it will cool the Seer is the efficiency higher seer less monthly electricity demands.
I have installed larger evaporators in the past when I knew the owner was going too small. The evaporators can handle a range of cooling where the compressor and condenser are usually 1 unit (the outside unit).
I would check with your installer is this unit have a dual speed or variable speed fan that is not being turned on? I have seen ac units air handlers on low speed and they did not cool as well. For a friend I installed a new controller and my friend thought I put a new system in it worked so much better (~150$ took care of him)
So there are possibly things that could improve the performance of your existing system insulation and possibly different fan speeds both of these would be much cheaper than a new compressor / condenser and possibly a evaporator if yours is not large enough to handle a bigger compressor.