I think tener que + an infinitive has come up quite a bit over tbe units and we are just supposed to pick it up. By the way, I don't think you'll see querer que as querer is followed by the straight infinitive. Thus: tengo que salir but quiero salir.
Context I believe. Tarea usually means task i think so a school-related task is your homework. Or if you prefer you can also use 'deberes'. General work in the sense of your job, or to say i am going to.... work is trabajo.
I' m not sure about "ahora" alone sometimes meaning "right now," but I do know that "ahora mismo" means "right now."
Well, putting aside the fact that they are two different translations in Spanish...
Both imply an obligation to do something however... technically 'have to' implies the obligation comes from someone else where as 'must' implies the obligation comes from the speaker.
My Spanish teacher taught me that it is la tarea (or los deberes in Spain) rather than mi tarea, unless you really need to emphasise the ownership (ie I'm doing my homework, not his)