Wise words above, and to start with, I have to admit I saw this when posted, but the term APU threw a curve ball at me, I had never heard that term before, so I shied away from responding at that time, more so as a search of APU showed nothing that made sense to me, so I figured it was something I have never seen, well it is really, but after seeing the above, and more searching I finally found out what it was about, but I had never heard the term before this.
Anyway now I know what it is, I can expand on the above, but the main things to note are, software readings are not always that accurate, and the first thing I would suggest is checking the temp reading in the bios, see how that compares to the other readings, but due to the amount of variables affecting temperature it can be a case of more than 1 remedy is required, so to start with these are my opinions on temperatures (others will have theirs, but all would agree when things start to get dangerous);
I would expect to see the temperature at around 86°F (30 degrees C), up to 104°F (40 degrees C) in some cases.
When in full use I wouldn't be concerned if it reached the 140°F (60 degrees C) range, and that's under heavy load, although I would be starting to look to get it down if possible.
BUT, when it goes past 158°F (70 degrees C) I would start looking to better it straight away, I'm not saying it would cause damage, yet, but it would be the precursor in my eyes, and systems I touch I set the warnings in the bios to start once it hit's this temperature.
When it hits 176°F (80 degrees C), I would start worrying about physical damage, although the temp wouldn't immediately do this, prolonged running at such a level will certainly shorten it's life.
The last one I will note is 194°F (90 degrees C) and above, this is the point I would start to say turn the system off to protect it (and it's what I set the bios to do), and in some cases it "may" have already done some damage, but please don't think this is a definite outcome, there are far too many variables, but needless to say the higher the temp once your beyond this point, and the longer it's left, the more likely it becomes you could fry your CPU, and possibly other components as well, but again these are the things I personally go with, others my go a bit further, and in rare cases like the P4 series, they were renowned for running at very high temps, which can make a difference to these temps I use.
Now you sound like you have a good idea about what I cover next, but as this may help others reading this with a similar issue, I will mention them to be safe, to start with the screws you mention, when they get to that stage, you are at risk of stripping the heads, which means you would need tools to sort it out, and if you can get them out, do NOT put them back in, get new screws, as if you do it too much you will need tools to remove the screw(s), and it can be very difficult depending on your luck, so rule of thumb is if your screwdriver doesn't fit very well when you first use it, get a better fitting one, it will save you a lot of heartache down the road, more so if you like upgrading etc.
The Push\Pull query (air flow) is one of try it and see, again due to all the variables it could make a big difference, other times there won't be much in it, I remember experimenting many years back with this, and I found little difference, but I personally have the fan blowing ONTO the heatsink, but you will come across aftermarket heatsink's with 2 fans, one is blowing on (push), the other one on the opposite side of the heatsink away (pull), this compliments the other and helps in that product.
The biggest thing that can make a difference is the case, it's hard to picture it unless you have dealt with it, I had a mate with very high temp's, and they had a small case, which due to size was quite cluttered, I was lucky to have a spare case, so all I did was swap their parts into my other case, I did not touch or do anything more, and the temp's dropped by over 86°F (30 degrees C), so the design and airflow inside the case has a massive impact on this, although with newer cases these days that has become less of a problem. but I mention it incase.
Heatsink, the stock one that comes with it is always good enough to start with, otherwise it would never be there, BUT, you can get far better temp's by using an aftermarket option (that's my findings anyway), but if you are not happy with your temp's, then this is usually a great solution if you get a quality part, but be aware the better ones are massive, and may not fit in small(er) cases, so be aware of this, but if you intend to max the system out then it's always a good idea to get a replacement for the stock cooler, otherwise if the temp's are constantly high, then this is usually the first thing to try.
That said, I do mean if the temp's were constantly high, if you have normally been used to low\reasonable temp's, and they suddenly go up, reseating the heatsink with new thermal compound is usually the first thing to try, and when doing this it's essential to clean off the old BEFORE applying the new, and nearly always sorts it out, but only when it has changed from what it has been running at since you got it, or if 2nd hand is to be considered not knowing it's past history, being the cheaper thing to try first.
Fan's\Airflow, no prizes for guessing the impact this can make, but often people underestimate this, and don't put the fans in the best places, or the best way round, so in simple terms all fans should be pushing the air toward the rear exhaust fan, if your case has a front fan (or space for one), this should be pulling the air into the case, if you have top fans they should be pulling the air out of the case, side fans vary IMO, some are made to pull air out of the case, more so ones that have a "funnel" that goes next to the heatsink I believe, other times it's experimenting.
Placement, no surprises for saying keep the tower\unit away from heat sources, such as radiators\fires, even direct sunlight which some may not think of, if you have bottom fans, or simply a mesh in the "floor" of the case, then if on carpet raise it up, and put say some hardboard (or anything flat and puts a barrier between the carpet and the intake, hardwood floors don't need this) on the floor and sit the tower on that, it reduces the dust\fibres from the carpet being sucked up into the system.
Cleaning, open the case every so often, say yearly, and give it a spring clean, when using a hoover to suck out the dust, if you are going close to the board it would be wise to use an old pair of tights, and stretch and cut a piece to place over the end of the hose, that way worst case, and a jumper or something loose won't get sucked up, you can just turn the hoover off and put it back, if using compressed air be careful how close you get to the board for the same reason, and is best done outside unless you want a dust cloud in your place.
A general note, when going inside a case ALWAYS practice safety, shut the system down, remove the mains cable completely from the system (and battery if a laptop), even turn off the switch on the PSU if it has one (belt and braces as they say), and ALWAYS touch the side of the case before touching the component's, it reduces the risk of static killing the parts, it only takes a couple of steps on a carpet to generate a charge enough to do damage.
Then depending on what your doing, take notes of wire colours and positions, take pictures (most people have a digital camera these days, even if only on their mobile), this way worst case, and something comes loose, you should be able to put it back safely and easily.
I've probably missed a few points, and sorry for writing a novel lol, but hopefully this will give you things to look at and check, and see if anything makes a difference, if it doesn't, then the only solution is to improve the cooling, the heatsink being the place to start IMO, so hope this helps a bit anyway.