Yes they can equip weapons and armor.
The do not inherently have disadvantage with any weapons or armor.
Source: Monster Manual pg. 272
Animated Dead. Whatever sinister force awakens a
skeleton infuses its bones with a dark vitality, adhering
joint to joint and reassembling dismantled limbs. This
energy motivates a skeleton to move and think in a
rudimentary fashion, though only as a pale imitation of
the way it behaved in life. An animated skeleton retains
no connection to its past, although resurrecting a
skeleton restores it body and soul, banishing the hateful
undead spirit that empowers it.
While most skeletons are the animated remains of
dead humans and other humanoids, skeletal undead can
be created from the bones of other creatures besides
humanoids, giving rise to a host of terrifying and
Portion that applies directly to the question, emphasis on important bits is my own:
Obedient Servants. Skeletons raised by spell
are bound to the will of their creator. They follow
orders to the letter, never questioning the tasks their
masters give them, regardless of the consequences.
Because of their literal interpretation of commands
and unwavering obedience, skeletons adapt poorly to
changing circumstances. They can't read, speak, emote,
or communicate in any way except to nod, shake their
heads, or point. Still, skeletons are able to accomplish a
variety of relatively complex tasks.
A skeleton can fight with weapons and wear armor,
can load and fire a catapult or ttebuchet, scale a siege
ladder, form a shield wall, or dump boiling oil. However,
it must receive careful instructions explaining how such
tasks are accomplished.
Although they lack the intellect they possessed in
life, skeletons aren't mindless. Rather than break its
limbs attempting to batter its way through an iron
door, a skeleton tries the handle first. If that doesn't
work, it searches for another way through or around
Yes, there is a direct reference to weapons. No, they are not at disadvantage.
Source: Monster Manual pg. 315, under the Mindless Soldiers paragraph.
A zombie armed
with a weapon uses it, but the zombie won't retrieve a
dropped weapon or other tool until told to do so.