Yes
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Hanuman_2000Hello,
1. I ________ (wrote/have written) the letter already.
As the word "already" is used with past tense and perfect tense. Both seem all right to me.
2. The boy _________ (brushed/has brushed) his teeth and went to sleep.
The words in bold suggest past tense so (brushed) is correct.
3.
CliveShe ________ (ran/has run) to catch. But the bus did not stop.Goot catch, Clive
'Catch' here requires an object. eg 'she ran to catch it'. This lack of an object makes me think this test question was not written by a native speaker.