Hi Gary,
I no longer live in Tetuán but do have my office there. When you say that the school you are looking at is on Bravo Murillo, do you mean where you plan on getting your CELTA? If so, that must be the BLC, which is practically on Plaza de Castilla. For the sake of this post, I am considering Tetuán the area covered by metro line 1 from Plaza Castilla in the north down to Cuatro Caminos, and the area covered by line 10 from Plaza Castilla south to Nuevos Ministerios. And from Nuevos Ministerios to Cuatro Caminos on line 6 (area between Calle Bravo Murillo, La Castellana and Raimundo Villaverde).
As an English teacher who has done her duty traveling the city to corporate clients on public transport, I always recommend living close to a centrally located line and quick access to one of the "corporate corridors"...in other words, quick access to transportation hubs that will get you to corporate clients along the N-6 (intercambiador in Moncloa), N-1 (intercambiador in Plaza Castilla) or N-2 (intercambiador in Avenida de America).
The center is a great place to "play". But I would rather live "close" to where I work instead of facing a 40 minute metro ride to get to one of the transport hubs to get out to a corporate client (another 45 minute ride) at 7am. Most corporate clients are along the N-1 and N-2. But that is just a personal preference of mine.
Regarding Tetuán, I think it's a great place to live. If you get a flat tucked away in a neighborhood (rather than on Bravo Murillo itself) you will find yourself immersed in Spanish life. It is a very multicultural neighborhood (then again so is most of the center of Madrid) with immigrant groups from Latin America (esp. Ecuador), Morocco and China. I like the variety of ethnic food shops in the neighborhood.
There are also many many longstanding Spanish owned businesses in the area. A great place to get churros is at Casa Aurelio on Bravo Murillo across from the Valeacederas metro stop. There is also a sizeable "polideportivo" (think YMCA) between Valdeacederas and Plaza Castilla and one of the best kept secrets, Plaza de la Remonta which has great terraces in the spring/summer/fall. And a great homemade ice cream shop! Closer to Cuatro Caminos you have "El Mercado de Maravillas" which is one of the larger and older markets in Madrid.
Areas such as those serviced by metro stops Plaza Castilla, Valdeacederas and Estrecho are less "masificado". Metro stops Cuatro Caminos, Alvarado and Tetuán are considered "little Latin America" due to the volume of Latin American immigrants.
Tetuán is a good area serviced by four metro lines (1, 6, 9 & 10). Lines 1 & 10 are good for traveling out the N-1 as they pass through Plaza Castilla. Line 6 will get you to Moncloa and line 6 and 9 will get you to Avenida de America. There are also a countless number of buses that pass through the area.
Prices for a one bedroom apartment will probably run you about 600-700€/month. A bit more in the area around Plaza Castilla and closer to La Castellana.
Bravo Murillo is a very commercial street with a lot of activity. It is closed off from Pza Castilla to Cuatro Caminos a couple of times a year on the weekends for one social activity or another (bike races, el día del niño, etc...).
If you have any other specific questions, just ask. Sorry for the long post... it's the start of a holiday weekend and my students have stood me up!
Good luck!