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Does Hinduism permit kings to act tyrannical? As in allowing kings to exploit the people, loot them, kill, take women, etc?

No, Hinduism forbids kings from acting tyrannical.

Some verses from the Manusmriti:

The king, who, without affording protection, takes tributes, taxes,
duties, presents and fines, would immediately sink into hell.—(8.307)

He who affords no protection and devours the people, grabbing his
tribute of the sixth part of the produce,—him they declare to be the
imbiber of the filth of the whole people.—(8.308)

He who heeds not the bounds of morality, who is a disbeliever, who is
extortionate, who does not afford protection, and is grabbing,—such a
king one should regard as doomed to perdition.—(8.309)

He who, on being abused by men in distress, forgives, becomes exalted
to heaven, by that act; while he who, through kingly pride, does not
forgive, goes, by that act, to hell.—(8.313)

So a king who is unrighteous and oppresses his people goes to hell.

Also, just like the US 2nd amendment, one can take up arms to defend himself and others, even against a tyrannical king, according to the Manusmriti:

Twice-born persons shall carry arms: When religion is interfered with,
when there is confusion among the twice-born castes caused by the
exigencies of time,—(348) in his own defence, in cases of hindrance of
sacrificial fees, in the case of outrages upon Brāhmaṇas and women,—if
one strikes in the cause of right, he incurs no sin.—(8.348-8.349)

And Medhatithi's commentary for that verse:

Another interpretation possible is that—“when religion is interfered
with, when there is confusion caused by exigencies of time, i.e., when
things have become unsettled on the death of a king—one may take up
arms; but at other tiroes the necessary protection would be afforded
by the king himself.”

But in reality the king cannot spread out his hands and reach every
individual person in the kingdom. There are some desperados who attack
even the boldest, and the most trusted officers of the king; but they
fear persons carrying arms.

For these reasons it is right that one should carry arms at all times.

‘Caused by the exigencies of time,’—such as the death of the king, and
such other calamities. On all these occasions one shall carry arms fur
the protection of his properly and family.

Others hold that on the occasions stated, arms may be carried for the
sake of other people also;—says Gautama (21.19)—‘Also when some one is
striking a weaker man, if he is able to rescue him.’

Commentary called Madanapārijāta:

if there is interference with the sacred duties due either to the
tendencies of the king or to the tendency of the times

Commentary called Mitākṣarā

when, on the waning of royal authority due to foreign invasion, one
has to take care of himself