So for polarity reversal causing no damage and requiring no fuse replacement you can use pretty much whatever diode you want and put it in series so that "normal" current flow passes through the diode only if properly plugged in. With the current requirements and voltages that you're working at, this shouldn't be an issue. A simple silicon diode should be fine.
For overvoltage you're going to want a circuit more like what Nick Alexeev suggested in the comment. Essentially a zener diode with a PTC or other type of fuse. The Zener should have a value which is less than the maximum input to your regulator.
So basically, if you reverse batt_in+ and batt_in- the first series diode will prevent any current from flowing and protect your circuit. If batt_in is greater than the breakdown voltage of the zener, it will start pulling down a lot of current, and blow the PTC fuse.
The only extra thing you might do, is to guarantee that the startup current doesn't exceed your PTC's current limit, you can place a resistor on "protected V_IN+" or "protected V_IN-" (in series before the regulator and decoupling capacitor) such that:
(BATT_IN+ - V_forward_diode - Resistor*Maximum_expected_load) >= Vmin_regulator
For the desirability of any specific characteristics for the PTC, the diodes, and everything else, it all depends on your application. In general, I tend to wing it unless I have a real reason to crunch the numbers. I'm also a bit too tired (on my way to bed) to really get into how to calculate what these values should be, but if you need this info ask in a comment and I'll post some tips on getting the numbers.
Though, why not just use a polarized connector for the batteries so that you don't have to worry about whether the connector is plugged in backwards? And in what context are you going to overvolt? Think about these questions too when trying to answer a more complicated design choice (a polarized connector is easier than adding an extra diode, and is less likely to lead to extra design considerations).
Hope that helps!
I recently read an article ... that seemed to indicate that "supercapacitors" are starting to approach the energy densities of batteries for some applications.
It's possible that there may be niche applications where that is true (although none come to mind with a quick musing) but for even 'ordinary everyday' batteries they have a way to go yet as regards either mass or energy densities.
BUT, as can be seen below, they have some utterly fantabulous specs that batteries cannot hope to match. eg 1,000,000 cycle life, 1000A+ max discharge current, 100A test current, ... !
Modern high end NimH batteries have energy densities about the same as typical LiIon cells. I'll use a typical NimH AA cell for comparison, but results would be similar for Lion or LiPo. LiFePO4 has perhaps half the energy density of LiIon but even LiFePO4 is far more energy dense that good supercaps.
An eg AA (14500) NimH cell weighs about 33g and provides about say 2500 mAh at 1.1V mean. That's conservative. Energy = 2.5 Ah x 3600 s/hr x 1.1V = 9900 Joule.
Say 10,000 Joule.
A capacitor discharged from Vmax to Vmax/4 delivers 15/16 of it's energy (as E= 0.5 x C x V^2).
So discharging a say 2.7V capacitor to about 0.675V uses most of the stored energy and is still a high enough voltage for operating a boost converter. A boost converter at around 0.6V has lower efficiency than at say >= 1 V but efficiency is liable to be acceptable if accessing stored Joules is more important than maximising efficiency.
E = 0.5 x C x V^2 x 15/16 = 9900 so
C = E x 2 x 16/15 /V^2 = 2897 F
Say ~= 2500 to 3000 F at 2.7V.
Digikey cheapest in that range are Maxell K2 series
2000 uF = 61mm dia x 102 mm long 360 g $55/1, $44/250
3000 uiF = 61mm dia x 138 mm long 510 g $60/1
AA Nimh = 14mm dia x 50mm long 33 g $3/1 ?
Cap - 1,000,000
NimH - 500
Short circuit current - Amp (also abs max for caps)
2000 F 1500 A
3000 F 1900 A
Nimh ... 10 A
Toperate C max/min
Cap +65 / -40
NimH -45 / 0
A 1,000,000 cycle life is quoted but temperature modified calendar life is liable top be the limiting factor. Data sheets for several brands claim 10 year lifetimes at 25 C with the usual Arrhenius equation effect of halving lifetime for each 10 degree C rise in operating temperature. If due care was not taken there are many locations where a 35c operating temperature could occur very easily, with a consequent 5 year typical lifetime.
There will be applications where forced air cooling and even heatsinking may be useful.
Here's a teardown of a satellite interfaced fishing buoy - Mikes electric stuff August 2014.
At this point: https://youtu.be/mY2X-ZQpnvY?t=475
You see this. Obviously the cost is irrelevant in the circumstances and the advantages outweigh the fact that this has about the same storage capacity as a good AA Nimh cell. There is space for a second one, but only one is fitted.
Since the product you purchased has no background information, you can't be certain. However, the convention for these stacked-disk type capacitors is polarity mark points to negative lead. This is the same as is the convention with conventional electrolytic capacitors.
For example, the Eaton KR-5R5V474-R:
Has its datasheet show:
Similarly, for the Panasonic EEC-S0HD224H:
Has the same arrow convention considering the asymmetric leads with polarity indicated in its datasheet: