You address 4 separate issues. First, an air admittance valve aka Studor, is legal in some areas and not in others. If an inspector recommended it, it must be OK where you are.
although most professional plumber don't like them, they do work, but must be install with access for replacement as they can fail over time. Second, the slow moving drain is a sign you may have a partly clogged line. Homeowner sized snakes are seldom effective on these. Neither are chemicals. Once you determine there is nothing caught in the toilet trap, the solution is a professional plumber who has an industrial sized auger and knowledge to use it to clear the drain. Your third problem has no remedy. Water level or water spot in the toilet bowl is determined by the weir that is the trap cast into the toilet, and there is no adjustment for that. Unfortunately, there are many toilets in use today that are really very poorly designed. What's even worse, many poorly designed toilets are still being sold, often under very well known and famous brand names. Do you need a new toilet? Hard to say until the drain system has been cleared and otherwise evaluated, but you should know that poor flushing and frequent clogs are not a price you have to pay to have a low flow toilet. For example, many Toto toilets are now using as little as 1.28 gallons per flush and they work very well. It's all in the design and quality control. Oh yeah, these toilets also have a generous water spot.