To be completely honest, there are plenty of myths and popular misconceptions about longbows and longbow-using archers, i.e. longbowmen. The whole "longbows are from Wales" thing is somewhat dubious. IIRC, the form of the longbow that became popular on the British Isles since at least the High Middle Ages might have been based on some longer Scandinavian bows used by the Scandinavian raiders and settlers back in the day. Another thing is not differentiating between different variations on the longbow (which is, after all, a bow like any other in its basic mechanics) and treating them all like warbows (the ones with military builds and poundage) from either the height of the Hundred Years War or the Tudor period. Longbows had variety, and post-16th century wooden longbows or modern day wooden longbows are not necessarily the same kind of weapon, but more of a sports version. Period types of hunting and military longbows were a bit different from each other, especially in terms of performance, and longbows evolved within each century, just like any period weapon.
Some good articles on longbows:
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Archery articles in general
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The Medieval English Longbow
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The Decline of the Longbow
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Some Speculation on the Nature of Longbowstrings
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Further Speculation on the Nature of Longbowstrings
Some good videos on longbows:
The Longbow - Mike Loades on the usage of the longbow in the Hundred Years War (including for mounted infantry hit-and-run attacks/raids)
English longbows vs. medieval plate armour (Battlefield Detectives documentary review) - Matt Easton criticquing a docu about the usage of longbows in the HYW (the docu seems to go for parroting a lot of the established myths/clichés)
Comparing the shots per minute of a longbow and hand-spanned light crossbow - a live test of
roughly how long it took to reload and shoot a longbow and a
purely hand-spanned crossbow during the same time limit, side by side
Sword and bow use TV stereotyping and strength - Matt Easton on stereotyping and inaccurate portrayals of an archer's strenghth on TV
Unusual late-medieval arming sword in the Wallace collection - a video on what longbowmen probably used among their melee weapons later in the HYW, in order to deal with the increasing usage of plate armour
Reasons for having the arrow on different sides of the bow - details are important
Military archery and speed shooting - Easton on why Rule of Cool isn't everything, especially not in military archery
Anyway, there's plenty of other medieval military resources listed
here, if you're interested.
For a horse (or a deer, or other game animal) you need a broadhead, which will slash in and sever nerves, muscles, and blood vessels. Chain mail could be penetrated by a bodkin - there's some disagreement as to whether modified versions had a blunter head in order to defeat plate armour, in much the same way as a kinetic AP round will be blunt in order to punch through armour without the risk of glancing off, or the thin point breaking on impact.
In terms of arrowhead types, both the bodkin and especially the broadhead are barely effective against plate armour. Bodkins have a better chance at getting through mail, but yes, there isn't a 100 % consensus on how they'd behave in every iteration of such a situation. It would be quite context-based. Broadheads, being the hunting arrowheads and general purpose arrowheads, would not be very effective against any metal armour.