Answer:
The answers to these fill-in-the-blank questions will depend entirely on context and gramatical structure. There is no real explanation as to why the word you will see is the correct one, except through maybe translation of the sentences. So, given this, the answers would be as follows:
1. Yo sé (know) cómo se llama: Valentina. Sé, is equivalent in English to knowledge (saber) or know. Translated, this sentence is: I know her name, or, I know what her name is
2. En un nombre bonito, ¿no? quiero saber, or conocer (both knowlege) más de ella, pero soy muy tímido. In this case saber, and conocer are used as synonymous and both point to ways of knowing a person.
3. Él conoce (knows) a casi todas las personas de mi programa. Conoce is another form of knowledge. This sentence would translate thus: He knows almost all the people in my program.
4. Pero si Julio no sabe (knows) quién es ella, mañana después de clase voy a escribirle un mensaje a Valentina. Sabe, is again, a form of knowledge, or how to get to know someone.
5. Tú conoces los poemas de Bécquer, ¿no? son muy románticos. Conoces is the plural of conocer and is again, a form of knowledge.
6. Yo conozco un café muy bueno que se llama La Rana. Conozco is another form of knowledge, a form of conocer.
