As long as he is not being summoned in the Damage Step, then yes, you can. After he is summoned, his triggered effect activates. You will chain Torrential Tribute to this.
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Triggers will always go on the chain first before any manually activated cards like Torrential Tribute. Black Rose Dragon's trigger will be chain link 1, Torrential Tribute will be chain link 2, so Torrential Tribute will resolve first, then Black Rose Dragon's effect.
When "Black Rose Dragon" is Syncro Summoned, the controller can choose to destroy all cards on the field. You can chain "Torrential Tribute" to the effect of "Black Rose Dragon", but the result is the same. The chain will resolve and "Torrential Tribute" will destroy all monsters on the field; but the effect of "Black Rose Dragon" will remain and the rest of the cards on the field will be destroyed. If you want to stop "Black Rose Dragon" from destroying all the cards on the field, "Black Horn of Heaven", Horn of Heaven", "Solemn Judgment", "Thunder King Rai-Oh" and "Royal Oppression" can negate it's Summoning and "Divine Wrath" can negate and destroy it when it tries to activate its effect.
Slifer's only protection is that spell, trap or monster effects cannot be used in response to his summon, meaning he is safe from cards that specifically respond to summons, such as Torrential Tribute. After that, targeted traps can be freely used against him.
You need to tribute 2 monsters to summon a Lv7 or more monster, so you need 2 for a 'Blue-Eyes White Dragon'.
Dragon type Pokemon are weak against Ice, Dragon, and Fairy type moves.