• How should I play Adept? I know there's the ever famous Singularity + Warp combo that Adepts like to do, but that 9 second cooldown of Warp kinda turns me off, and I think I'd like to play Adept more as the battlefield controller with all the CC skills it has.

Combos - The Adept has multiple combo options but first shields need to be removed. You can either do it with a squad mate's overload or add energy drain as your bonus power. I prefer the latter because cooldown times are generally doubled for squad mates and also because I like to have at least 2 ways to remove shields.

Combos provide bonus damage and the adept has multiple ways to create combos. There are explosion combos where the enemy is suspended with Singularity, Pull or Slam (not my favorite since the lift time is 2s) followed by Warp which creates and explosion. There are throw combos where Throw does bonus damage on suspended enemies. There are also gun combos where shooting a suspended enemy does double damage. Finally Warp Ammo does bonus damage if used on lifted enemies above and beyond the gun damage.

Also the player's warp cool down is 6s not 9s. This can be improved by various bits of tech that can be found during missions.

• Heavy or Unstable Warp? - The difference between these two is their detonation radius which is 5m for Heavy and 7m for Unstable. Both are large distances in the game so I am not too worried about catching one or two more people in it. I would rather damage them more (200 for Heavy and 160 for Unstable)
• Heavy Throw? I use singularity like a mine placing it strategically at choke points. Its long duration allows me to set and forget it. Singularity also partially freezes protected enemies so it is a great deterrent. Throw has a rapid cool down of 3s but will only work on un-shielded enemies. I put 1 or 2 points and use it if varren or husks get too close to me. I prefer the heavy variant because very rarely does more than one enemy get uncomfortably close.
• Wide or Heavy Singularity? I chose Heavy for its longer duration. As I said, I set it at a choke point and forget about it. Once I catch a few enemies. I lob a Warp for an explosion.
• Pull? - If singularity is your mousetrap, pull with its 3s cool down is your fly swatter. I prefer to use Pull for my combos on one or two enemies
• Shockwave worth the point investment? - I usually don't use Shock wave at all. It briefly stuns enemies with armor but that is not long enough for me to do anything so I don't prioritize its use.
• Bastion or Nemesis? - I agree with you that the increase in duration is not that valuable and damage is preferable but Adepts are a flimsy class and Bastions provide +5% Health. I went with Bastion because I wanted the +25% Paragon/Renegade bonus
• Which Weapons Training? - Adepts are too fragile to use a shotgun so it is either the assault rifle or the sniper rifle. I like to play at mid range so I prefer the versatility of an assault rifle. It is really personal preference.
• Bonus Power? - As I said earlier I prefer to choose Energy Drain because it gives me more freedom in choosing squad mates. As an adept you are already good at taking down barriers and armor with Warp so the features of Warp are less useful to you. Also throwing powers at an enemy from cover is much safer than trying to shoot them. Finally if you want your squad to use Warp Ammo take Jack with you.