Я сушу одежду на балконе.

Russian prepositions for location follow certain rules:

• на
  • prepositional case is used for surfaces or open spaces (e.g. на столе, на улице, на балконе).
• в
  • prepositional case denotes being inside an enclosed space (e.g. в комнате, в шкафу).
    A balcony is treated like an open surface rather than an “inside” space, hence на балконе with the prepositional ending -е.

Сушу is the present-tense form of the imperfective verb сушить (“to dry”). Imperfective verbs:

• Describe ongoing, habitual, or repeated actions (“I am drying,” “I usually dry”).
• You cannot use them in the future single (simple) form; instead, you’d say буду сушить (“I will be drying”).
  The perfective counterpart is высушить (“to dry completely”) or подсушить (“to dry partially”). Perfective verbs focus on the completion of an action and form a simple future (я высушу – “I will dry [it]”).

Here are the present-tense forms of сушить (stress is on the first syllable in all forms except the infinitive):

• я су́шу
• ты су́шишь
• он/она су́шит
• мы су́шим
• вы су́шите
• они су́шат

Yes. People sometimes say:

• развесить одежду на балконе – to hang out clothes to dry
• вывесить бельё – to hang laundry (бельё)
  Both emphasize the hanging action; сушить focuses on the drying process itself.