No, exhausts don't need to be removed, the bike should have 'open ended' chain adjuster blocks fitted, so you just pull the wheel out from the back.
Edit, just looked at your pic and you have the correct adjusters fitted, so just undo the adjuster bolts, the adjuster will drop down, remove the blocks and pull the wheel out backwards.
Another Edit 😁😁😁You won't be able to get a socket in to undo the spindle bolt nut, just use a spanner.