"I am walking and thinking."
Translation:Я хожу и думаю.
Not strictly incorrect, but they're usually avoided, especially in writing or formal speech. It's not a grammatical mistake, but a stylistic mistake: usually such sentences sound too colloquial.
This is related to the past tense. Since in the past tense you can't tell the subject if it's not indicated with a pronoun (e.g. «я ходи́л(а)» 'I was walking', «ты ходи́л(а)» 'you were walking', «он(а) ходи́л(а)» '(s)he was walking'), a subject is required in the Past tense. In other tenses, the pronoun is used by analogy.
Many Slavic languages allow dropping the pronoun. This is because these languages didn't drop the linking verb 'to be', like Russian did (e.g. in Slovenian you don't need jaz in (jaz) hodil(a) sem 'I was walking', because it uses the verb sem). Russian is different from other Slavic languages in this respect.