If foreign interest rates rise, this will cause the net capital
outflow to increase, but it will also cause domestic interest rates to
increase. Since domestic interest rates cause the net capital outflow
to decrease, what will really happen to the net capital outflow: will
it increase or decrease?

A lot depends on market sentiment.

If the market expects the USD to fall, then people will move into
other currencies regardless of the interest rate.

Since governments tend to increase interest rates to /prevent/ their
currency falling, a rise in foreign interest rates could be seen as a
sign that the foreign currency is in trouble.

Of course one can hedge against an anticipated fall in a currency, but
the cost of doing so needs to be taken into account.

Theoretically rises in interest rates are supposed to dampen down
consumer expenditure, but in reality, unless the rise is enormous,
consumers are not that sensitive to an increase
- generally they are paying substantially more than the 'bank rate'

Interest rate movements nowadays tend to be 'a sign to the market'
- what the market reads from the 'sign' is a different matter

As you are probably beginning to realize, Economics is a load of
mechanistic principles - which are outbalanced by psychology.