If I activate Firewall Dragon to bounce a card and the activation is negated, would I be able to activate Firewall Dragon's effect to bounce again or would it not be able to activate?
"You can only use each effect of "Firewall Dragon" once per turn." So, no.
Only once per turn
The more common type, found on Monster cards, and multiple use/off-field effects of Spell and Trap Cards. If a card states this type of restriction, and either its activation or effect is negated, you cannot activate it once again.
... with the difference between "once per turn" and "only once per turn" being "even if another copy of Firewall Dragon were or were to become on the field, the other copy could not activate this effect either".
Note: very new player, reading the rules and listening to people who talk about Hard Once Per Turn restrictions. IC Very Easily BW.