2. Factory Nav System - Let's take a reality pill here. All factory installed Nav systems that I've ever seen are overpriced and typically under-deliver (compared to aftermarket options). You have two choices: Live with it and operate within it's functionality.....or replace it with a significantly better aftermarket add on. Period. Those are your options. Complaining about what it can't do isn't going to help. Comparing it to a "better" aftermarket solution isn't going to help. I personally miss the days where I could "opt-out" of a factory radio altogether and just have my vehicle delivered with a $500.00+ additional discount that I could then apply toward a superior aftermarket unit. Unfortunately, there are NO car manufacturers that I know of who are doing that anymore. In reading countless posts on the factory system here on this forum, I've noticed that most of the functionality issues tend to be in many cases related to the device being paired with the Nav system. You might consider searching and reading to see if anything jumps out as a possible match for your issues. I have the factory (non nav) system in mine, and I find that it functions just fine as a radio and simple head unit for an ipod or usb drive with some music. It also works perfectly for me as a hands free interface to my android phone.....so I have no need to change it out.
3. Your TPMS issue - This should be solvable by your dealer. Tip: If your dealer hasn't been able to solve it in multiple visits, you might consider bringing it to another dealer with better techs. At the end of the day there are a finite number of components involved that could be the issue. Either you're driving on low tires (doubtful), or you have a sensor or part or the computer interface infrastructure that is faulty and needs to be replaced.
These 2 points are right on the money. #2 serves to showcase the basic limitations of the stock navigation unit. The bottom line is that it sucks. Use any free GPS/map service on your cell phone for better navigation. If there are interface issues on the unit itself, those are different and should be addressed by Toyota at some point.
#3 serves to highlight that it sounds like you might be dealing with idiots at the service center. Really, there are only a handful of things that could be wrong with the TPMS system. Their job is to find which sensor (or multiple?) is causing the issues you are having.
Unrelated to this, but on the point of spare TPMS since it was already mentioned..... the spare does not have a TPMS sensor in it and is not read by the computer. It could be inflated fully, flat, or removed completely and the computer would not know. At least, the 2010-2013 didn't have a spare TPMS, so I'm assuming this trend continued for 2014-2015 MY as well.