The -te form is used to connect two or more clauses that are somehow related to each other. The clause that follows each -te verb can be a single verb (as is the case in -te iru), or be longer (as is the case in 焼いて ... 上がる).
You cannot connect two verbs by simply putting them next to each other unchanged like "思う書く"; that's grammatically invalid. You have to use specific grammar to connect them, the most common of which is putting all verbs to be connected except for the last one in the -te form: Xて Yて Zる. The last verb then determines the tense of the complete list (past or present).
思っている would be present continuous ("thinking"). However, 思って書く means simply "think and write" (or better: "write with the thought of ..."), i.e. *not* present continuous. For "writing with the thought of", it would have to be 思って書いている.