Loose-leaf tea: specifics of re-steeping multiple times?
As you've noted in your description of your experiments, there are many variables (e.g., amount of tea, type of tea, steeping times for each infusion, etc.). Fundamentally, the type (green/oolong/pu-erh/etc., quality, nature) of tea is a huge factor, and personal preference is a factor but not the only factor (what you describe as "bland" or weak might not be to someone else). I think there are some objective generalizations, though.

The tea you start with matters. A specialty tea dealer will often annotate a given offering with brewing recommendations (e.g., commonly temperature, time, quantity of leaves per cup, as you note); some dealers will also suggest that the tea can be enjoyed through 2-3 or multiple infusions. Such a tea is a great place to start for your own experiments.  I find that some teas (e.g., darjeeling, broken orange pekoe) simply infuse too quickly and to fully in the first infusion that they are totally spent. Further infusions are weak and bitter.

Some teas that I find work well for multiple infusions:


good quality green-oolongs (multiple infusions; tung ting or formosa)
green-white blends like white peony (2 infusions)
lightly roasted/toasted green teas, like long jing (dragon well) or gunpowder green (aside: this is probably a good place to start as it can be inexpensive and enjoyable)
higher quality pu-erh teas.


Techniques. This varies per the type of tea; but in general, I personally do the following when I intend to infuse loose tea multiple times:


Infuse slightly more tea (perhaps 50% (1.5 times) more; certainly not 2-3 times);
Infuse for a shorter period of time (perhaps 1/2 to 2/3 as long) on the first infusion;
Infuse for progressively longer amounts of time for subsequent infusions.
If anything, slightly lower temperature on subsequent infusions.


That tends to make the resulting brew (to my taste) slightly more uniform in result, though there is significant difference in flavour.

Other things to keep in mind: if you're using tea with caffeine, most will come out in the first infusion; if you're using 2-3x the tea, this is significant! As you also note, the character of the brew can be very different through the several infusions. This can be part of the enjoyment, as each infusion will extract different ratios of "stuff" into the resulting brew. For lower-quality gunpowder green teas, for instance, I actually prefer the second brew over the first.

Instead of multiple infusions, you might also simply


consider brewing a larger pot all at once into a thermal carafe, then enjoying it a cup at a time.
consider brewing just a cup at a time, which will have a better result for certain teas (a pathological example: bags)


I would not recommend some things that you said: most specifically, I don't think you'll get benefit from infusing at a higher temperature for subsequent brews; this is likely to give you bitter or off-tastes; you're "burning" the tea and extracting stuff you don't want.