OK, so when $b$ is high, we basically want to bet $p$ of our bankroll. This is easy to remember. I would like to think of the Kelly bet as kind of an adjustment on that.

Given fixed $b$, the Kelly bet is a line from 0% at the break-even point to 100% on sure things. So where is the break-even point?

The break-even point is where the top of the formula equals zero, ie $p(b+1)-1=0$, so $p=\frac{1}{b+1}$. I prefer to think in terms of $r$, the "returns", $r=b+1$. (If there's an opportunity to "double" your money, $r=2$ while $b=1$; so I think $r$ is more how people intuitively think about things.) So the break-even is $p=\frac{1}{r}$. That's also pretty easy to remember.

So we can just check whether $p$ is above $\frac{1}{r}$, which is a little easier than calculating the expected value of the bet to check that it's above zero.

As noted earlier: when $b$ is really high, we bet $p$, sliding from 0 to 1 as $p$ slides from 0 to 1. We can now state the adjustment: when $b$ is not so high, the Kelly formula adjusts things by making the slide start at $p=\frac{1}{r}$ instead of at zero.

So one way to calculate Kelly is to ask how much along the way between $\frac{1}{r}$ and $1$ we are.

For example, if $r=2$, then things start to be profitable after $p=50\%$. If $p=80\%$, we're 3/5ths of the way, so would invest that part of our bankroll.

If $r=3$, then things start to be profitable after $p=33.\overline{3}\%$. If $p=66.\overline{6}\%$, then it's half way to $100\%$, so we'd spend half of our bankroll.

This seems way easier than trying to mentally calculate "expected net winnings over net winnings if you win" [LW(p) · GW(p)], even though it's the same formula.

Rather than talking about p and b (your probability and net fractional odds*) doing things in terms of p and q = 1/(b+1) (your probability and implied odds probability) makes calculations easier and lots of the things you didn't expect more intuitive. Specifically:

$$f=max(\frac{p-q}{1-q},0)$$

This means:

1. It is easy to see the fraction is piecewise linear
2. The answer to Q5 is obvious.

* aside: who talks in net fractional odds? None of the 3 major odds styles use it

I am interested in a couple of points around this:

What makes this worse is that the region of 0.05% to 1% of my net worth is full of long tails. The wagers I'm skipping could easily repay themselves a thousandfold. If I take a wager like this every day for 2 years and just a single one of them repays itself a thousandfold then I win bigtime.

If I take b = 1000 and f* at 0.0005 I get p ~= 0.0015

If I take b = 1000 and f* at 0.01 I get p ~= 0.011

You draw from this the lesson that you need to bet more. What do you see as a source of potential wagers, coming along daily, where there's a potential 1000x payout and your probability of a win is between around 0.15% and 1.1%?

Also, I'd urge caution over the "[if] just a single one of them repays itself a thousandfold then I win bigtime" framing.

At the p=1.1% / bet=1% of wealth situation, one win in 730 (daily for 2 years) would take a starting balance of $100,000 down to $718. Now, treating this as binomial distribution, that'll only happen 0.25% of the time, but 2 wins leaving you with $7,990 will happen 1% of the time and 3 wins, leaving $88,871 will happen 2.8% of the time (assuming I've not mangled my calculations).

(Yes, in ~96% of cases you'd be left with $988,451 or greater, typically much greater, in the $millions, $billions or $trillions. But this is a caution against thinking 'only one of these 1% chances in 2 years has to come off to be winning bigtime'.)

At the other end of the spectrum you mention, i.e. the bets are all at the 0.05% of wealth (p=0.15%) situation, the results are much less dramatic.

wins=0, closing balance = $69,337 (prob = 0.33)

wins=1, closing balance = $104,162 (prob = 0.37)

wins=2, closing balance = $156,478 (prob = 0.20)

wins=3, closing balance = $235,070 (prob = 0.07)

wins=4, closing balance = $353,134 (prob = 0.02)

Note that the formula listed in the article is the Kelly formula for when you lose 100% of your stake if you lose the bet, which isn't always the case.

The Kelly formula is derived from the starting point of:

$bankrol{l}_n=(1+b\ast wager{)}^{pn}\ast (1-a\ast wager{)}^{(1-p)n}$

Essentially, after a sufficiently large number of n wagers, you expect to have won pn times and lost (1-p)n times. Each time, your previous bankroll is multiplied, either by (1 + b*wager) if you won, or by (1 - a*wager) if you lost. 

Often, a = 1. Sports betting, poker tournament, etc - if you lose your bet, you lose your entire wager. 

Sometimes, though it isn't: For something like an investment with a stop-loss, for example, the downside risk could be something like 20% instead of 100%.

If you leave it as "a" instead of assuming a=1, you end up dividing that first term by it:

$wager=\frac{p}{a}-\frac{1-p}{b}$

You say:

The Kelly wager is positive iff the expected value $bp-b(1-p)$ is positive.

I think this is supposed to be $bp-(1-p)$. 

The Kelly Criterion maximizes the growth of your bankroll over time. This is probably not actually the goal that you personally have for wealth, because of the nonlinearity of money. You (if you're like everyone else) care much more about preserving wealth, once you have some, than you do about growing it.

Some of this might be loss aversion, but mostly this is right -- going from $1M to $2M is nice but far from a doubling in your happiness or ability to do things; going from $1M to zero is a disaster. Kelly doesn't take that into account, except in the purely mathematical way that if you literally go to zero you can't make any more bets (which never happens).

For this reason, professional gamblers I know tend to bet half-Kelly to balance out bankroll preservation with growth. (Source: used to be a pro poker player.)

On the flipside, if you have another source of income, you can bet more aggressively. For instance, if you have a job that generates positive savings, you can count unearned savings as part of your bankroll for Kelly purposes. This is a huge advantage pure pro gamblers don't have. You probably don't want to be too too aggressive there, and how much to count will depend on the stability and/or fungibility of your income. A year or two of savings could be appropriate.

None of this should change your bottom line that you should take +EV longshot bets if you've been passing on them, just how much you should bet.

Does considering $b$ to be a kind of "misfortune reduction factor" help your intuition at all?

In an even-money bet ($b=1$), $f^{\star }$ is the edge: $f^{\star }=p-q$.  No reduction of misfortune here.

For general $b$, $f^{\star }$ is the edge with misfortune $q$ reduced by a factor of $b$: $f^{\star }=p-q/b$.

The same idea works for bets in which you forfeit a fraction $a$ (not necessarily $1$) of you bet upon losing: in that case, your good fortune $p$ is effectively reduced by a factor of $a$: $f^{\star }=\frac{p}{a}-\frac{q}{b}$.