Will continent bonuses accumulate

Yes, you can disable the tutorial by unchecking the tutorial box on the start game screen. It's in the box of settings below the difficulty selection.

Requirements to get Continent Bonus (from the XCOM wiki):

  |--------------------------------------------------------------|
  | Region           |  # of contact areas  |  # of radio relays |
  |--------------------------------------------------------------|
  | Africa           | Contact 3 regions    |  2 radio relays    |
  | Asia             | Contact 4 regions    |  2 radio relays    |
  | Europe           | Contact 2 regions    |  1 radio relay     |
  | North America    | Contact 3 regions    |  2 radio relays    |
  | South America    | Contact 2 regions    |  1 radio relay     |
  | Oceania          | Contact 2 regions    |  1 radio relay     |
  |--------------------------------------------------------------|

This is fixed no matter what difficulty you play on or what campaign you play on. The only thing that changes through different playthroughs are the specific Continent Bonuses that you can get, per BlueRaja.

The list of possible Continent Bonuses are:

• Quid Pro Quo
  • Black Market items cost 33% less intel.
• Under the Table
  • Black Market pays 50% more Supplies for all items.
• Lock and Load
  • Personal Combat Sims and Weapon Upgrades may be reused.
• Pursuit of Knowledge
  • Each Laboratory increasess research speed by 20%
• Future Combat
  • All GTS Tactics costs reduced by 50%.
• Armed to the Teeth
  • Extra mod slot for weapons.
• Spy Ring
  • +25% for Intel rewards.
• Hidden Reserves
  • Extra Avenger power output (+4).
• All In
  • Supplies from Resistance drops increased by 20%.
• To Serve Mankind
  • Recruits cost 10 supplies.
• Suit Up
  • Proving Ground armour and vest projects are completed instantly.
• Spare Parts
  • All proving grounds projects cost 50% less
• Fire When Ready
  • Experimental ammo, grenade, heavy and powered weapon projects are completed instantly.