Assuming that you are able to apply both a +5 Amulet and Bane to a single weapon, then that single weapon would have a +7 Enhancement Bonus against the designated creature type.
For the enhancement bonus limitation, it depends on the specific effect or ability that's altering the weapon.
Bane: This allows the weapon to exceed the +5 limit, but only against the designated creature type. For example, a +5 dragon-bane longsword is normally a +5 weapon, but has a +7 enhancement bonus against dragons and deals +2d6 points of damage against dragons.
Paladin: The divine bond ability says "These [enhancement] bonuses can be added to the weapon, stacking with existing weapon bonuses to a maximum of +5." That means if a paladin has a +5 longsword, she can't use her divine bond to increate the enhancement bonus to +6 or higher (but she could use her bonuses to add abilities such as flaming to the weapon).
The +10 bonus-equivalent limitation is a hard cap for all weapons; you can't exceed that even with class abilities or other unusual abilities.
Applying both could be tricky in this case since the Amulet only works on Natural Weapons and Unarmed Strikes while the Baldric requires that you hang the weapon from it for 24 hours if you are not an Inquisitor. That would require a manufactured weapon. If you are an Inquisitor the Baldric is easier to use, but would still be limited to one weapon. So If you had a claw/claw/bite all three would get the +5 from the Amulet, but only one would also get Bane.
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Normally I'd say that unarmed strike counts as a weapon, but I don't see how you'd hang it from the cloth for 24 hours....
Ever break your arm? You hang your arm in a sling. Cannot the baldric be used as a makeshift sling for a day?
/cevah
Does the unarmed strike qualify as a light or one handed melee weapon the baldric requires it to be?
Unarmed Strike is always considered a light weapon, so yes.