Hello,
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. She __________ (ran/has run) to catch. But the bus did not stop.
This also suggests past tense. so (ran) is ok.
Please check my answers.