Flouridated toothpastes are poisonous but as with anything our good friend Paracelsus comes into play here:
All things are poison, and nothing is without poison, the dosage alone makes it so a thing is not a poison.
By spitting the excess toothpaste out you are simply avoiding ingesting it in sufficient quantities to cause adverse effects.
The advice not to rinse is to allow the fluoride that remains in the mouth to do what it's there for - helping prevent tooth decay, yes you will end up ingesting small amounts of the toothpaste that remains but it's much, much, smaller than the quantity you would have ingested by swallowing the toothpaste rather than spitting it out. As per Crawford et al the lowest dosage of fluoride seen producing symptoms was 1.6 mg/kg (the mean amount in asymptomatic patients was much higher at 3.89 mg/kg) - the classic "pea-sized" lump of toothpaste is ~ 0.25g so taking me as an example..
I weigh 59kg, meaning that using the lower 1.6mg/kg figure I'd have to be ingesting 0.0944g of fluoride before symptoms were even a possibility - looking at a relatively typical fluoridated toothpaste it has 1450ppm of Sodium Fluoride which means our initial 0.25g of toothpaste contains about 0.0003625g of sodium fluoride. Which is 0.145% of that initial 0.25g! So even if all of the initial 0.25g were ingested it doesn't even come close to the levels needed to cause symptoms
So it's not sounding as though that small an amount stands much chance of causing a problem. If we look at the other end of the scale the mean dosage of those who were symptomatic is 4.78 mg/kg so at my weight that would mean ingesting 0.28202g of sodium fluoride, which at 1450 mg/kg would mean eating a frankly insane amount of toothpaste! So I won't lose any sleep over the negligible amounts I may ingest from not rinsing, if I do rinse however I'm going to be removing the opportunity for the fluoride to do it's work on the teeth and removing a known benefit for avoiding an essentially negligible risk isn't a sensible trade-off.
So coming back to your three questions:
Even after spitting, wouldn't some toothpaste linger in the mouth?
Yes, that's the point.
After brushing, it's obviously impractical to keep returning to a sink to spit every time you have saliva. So wouldn't you swallow the residual toothpaste?
A tiny bit yes.
If 2 is true, isn't swallowing any amount of toothpaste unhealthy?
Nope. The dose makes the poison and the dose is insufficient to cause health problems.