A fifteenth is, as the term suggests, a tax of 1/15th. Shakespeare uses the term to comic effect in Henry VI Part II 4.7:17-20 (see here). This case law book specifies that a fifteenth is levied on movable property, but I would not trust its accuracy for all uses, particularly since the OED gives:

A fifteenth part; esp. in Eng. Hist. A tax of one-fifteenth formerly imposed on personal property.

The "setting out of soldiers" is similarly a tax used to raise and equip an armed force.

The fifteenth - here fifteen - was an occasional tax from the 1330s to the 1620s, initially supposed to be a fifteenth of the value of non-exempt movable goods (a tenth in boroughs), but later levied as a conventional assessment by locality, its value eroding with the inflation of the 16th & early 17th centuries. Those with less than 5s of goods, later 10s (I seem to recall seeing a possibly still later £1 too) were exempt, so the poor referred to are presumably those just above the threshold.

"Setting out of soldiers" is a new one on me, presumably likewise a tax for particular spending needs, in this case explicitly military. It might have been a short-lived Elizabethan levy.

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Hmm. I'm wondering if the custom of tipping 15%, common in the USA, is a descendant of this tax?