Can all enemies be suppressed? For example, can you successfully suppress a Cryssalid or will they simply take the hit point and destroy you?
To the best of my knowledge, all ranged attackers can be suppressed, but melee attacks cannot be suppressed...in fact, aliens' melee attacks never miss under any circumstance.
I'd also like to note that I don't recommend taking the heavy's suppression skill at all, or at least not until you've completed the Ammo Conservation foundry project. I'd much rather have a shredder rocket for panic situations when the squad needs to focus something down fast than a skill that wastes two shots from the heavy's small clip and doesn't kill anything.
It should be noted that suppressing a melee alien will still grant the reaction shot on moving, but at a cost of two ammunition. Note, however, that you could just use overwatch at zero ammunition cost.
Since melee aliens always move unless directly adjacent to a unit, if the alien is in cover, preparing for a reaction shot may grant a better chance to hit than taking a direct shot. Since melee aliens rarely seek cover, in most cases, the 15% penalty to aim on reaction shots will mean it's less likely to connect than a normal shot taken on your turn, so it's usually best to just take your shots.
If you've taken other heavy skills that enhance suppression, you could gain a slight advantage from also suppressing other aliens nearby (granted by Danger Zone), and/or the two points of guaranteed damage to suppressed aliens (granted by Mayhem).
Dealing with melee aliens
There are only three ways to prevent a melee attack: avoid the alien, kill the alien, or stun the alien.
Since melee aliens run faster than XCOM soldiers, running away is usually not very effective, but may occasionally be the right choice, if the alien is still at a moderate distance, and is unlikely to be killed on the current turn. More often than not, however, it's kill this turn or be mangled next turn.
The Chitin Plating accessory can greatly reduce damage from melee aliens, so you might consider giving that to your forward scout if you find you're often losing your scout to melee aliens.
If you scout with an assault soldier you could use also the skill Close Combat Specialist to defend yourself against an approaching Cryssalid or Muton Berserker, especially if you have an alloy cannon equipped.