Yes the range is doubled
Note the range on Booming Blade is 5 feet, so following Spell Sniper it should be doubled to 10 feet.
Jeremy Crawford has tweeted on this topic, about a similar spell Green Flame Blade.
Q: What if he had the Spell Sniper feat? The spell requires an attack roll, would its range thus be 10ft, allowing the whip?
A: Spell Sniper does work green-flame blade.
No you cannot make an opportunity attack with a spell…
Booming Blade requires the Cast a Spell action, and only has an attack as one of the components. Opportunity attacks must be attacks, not spells.
To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature.
… Unless you have the Warcaster feat
From the feat:
When a hostile creature’s movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.
Since Booming Blade is a spell with a cast time of one action, and only targets one creature, you could cast the spell as your Opportunity Attack with the Warcaster feat.
Polearm Master adds additional chances to Opportunity Attack
While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.
Combining Warcaster, Polearm Master, and Spell Sniper would allow you to use Booming Blade at 10 foot range as a creature enters your reach.
Note that even though Booming Blade’s range is increased to 10 feet, since it requires a melee attack, your weapon must still reach that far to use it at this range.