So, the way combos in ME 2 work is that they have two components, a primer, and a detonator.
Whenever a target currently affected by a primer is hit by an eligible detonator, there's a chance that explosion occurs based on the specific combination of effects in play. For example, Biotic Charge is capable of detonating all four varieties of combos. If a target is hit by Warp first, which acts as a primer, a Biotic Combo will trigger, which explode in an area effect for decent damage against health, barriers and armor. Biotic Combos are also notable in that they are the only Combo that triggers 100% reliably, regardless of the health of the target. Others may require a killing blow, or a fair amount of luck.
If instead of charging a target affected by warp, we had charged a target that had been set on fire by Incendiary Ammo, we'd have a chance of triggering a Flame Explosion, which will set victims on fire, causing organics to panic.
There's a whole variety of combinations that can produce one of four types of combos (Biotic, Flame, Electric, Cryo), so experiment!
With regard to the various Detonation upgrades, they specifically effect combos detonated by that power. As a result, they are incapable of stacking.
There are definite synergies between various classes and certain weapon types. First of all, many powers and abilities only work with specific weapons, or give special bonuses when used with specific weapons. For instance, Vanguards have access to a number of bonuses to Shotgun damage, and Soldiers have access to several Assault Rifle upgrades. Infiltrators have access to lots of Sniper Rifle synergies.
Additionally, only Soldiers and Infiltrators experience the 'time dilation' effect when looking down the scope of a Sniper Rifle - this is invaluable for setting up headshots.
Finally, different classes have different weight allowances. Because some weapon types - notably Assault Rifles and Sniper Rifles - are much heavier than others, trying to carry one with a class with a low weight capacity (like an Adept) will result in a large penalty to your power recharge times. This is an important tradeoff, since these classes essentially live and die by effective use of their powers.
TL;DR: While you aren't required to use certain weapon types, there are definite rewards for specializing in something suited to the combat style of your class.
Best Answer
DoT effect of warp stays on target after detonation via Biotic Explosions. I got "warp" kills with asari adept after target survives Biotic Explosions.
Seems like Expose stays too. As long as i can judge damage outputs from pistol on Brutes that survived Biotic Explosions is bigger than on non exposed to warp Brutes. But there is always probability that this is effect from warp itself not from Expose Upgrade.
To carefully asses situation we need make an experiment on shielded non-armored target that can survive BE.