Once you start getting into the stronger spells, you will likely run into mana issues unless you have some sort of strategy for boosting mana. Otherwise, you'll have to pace yourself.
Acid cloud and Locust Swarm are two spells I use fairly frequently. If I open with Grasp of the Dead, then cast two Acid Clouds, I won't have enough mana for Locust Swarm.
It really depends on your build, though. Many pet-centric builds may not really require much in the way of mana assistance, since minions are "fire and forget" abilities.
I just ran a few tests because I was curious on this and wanted to know myself since I'm maining a Witch Doctor. My findings are as follows.
Testing Method: I used the abilities on a monster (or multiple) and watched damage against them. I removed gear and equipped it while DoTs were ticking on targets, and I also used INT Boost abilities while DoT effects were ticking on targets. To test Spiders I did something similar but had to time it better. I also had no runes active and no passives active. I didn't try to get screenshots, but if you'd like me to redo the test and try to get good screenshots let me know and I'll edit them in.
Scenario 1: Grasp of the Dead - Gaining INT while Grasp of the Dead is already on the ground does not appear to effect Grasp of the Dead's damage. Damage is calculated when spell is cast and damage dealt doesn't appear to change when INT is increased post cast.
Scenario 2: Corpse Spiders (already thrown) - Gaining INT while Spiders are already on the ground does not appear to effect Corpse Spiders' Damage. This was also true for Tiki Men.
Scenario 3: Haunt and Locust Swarm - Gaining INT while Haunt and Locust Swarm are already on a target does not appear to effect Haunt and Locust Swarm's Damage. Additionally, the damage of Locust Swarm bounces ALSO does NOT appear to be effect by INT gains.
Scenario 4: Summoned Pets/Minions - As far as damage is concerned: Damage is calculated for Gargantuan ON HIT. Meaning when he swings at a mob your stats at that moment are the deciding factor on how much he hits. This was also true for Zombie Dogs. I was not able to test if and how INT effects their other stats because I don't know a way to display that information for summoned minions.
Additionally, I also tested if weapon damage being applied changes the situations and all of the above scenarios are still true.
These tests show the following:
1) Damage calculation for persistent effects (DoTs, AoE Damage Pools, Non-Minion Summoned Creatures, and etc) is calculated as soon as the spell is cast and damage is not retroactively applied when the player's stats increase.
2) Damage calculations for minions (Gragantuan and Zombie Dogs) is calculated when that minion hits a creature, thus will gain bonuses for increased INT.
These were all tested on a level 60 Witch Doctor so I can't say if they apply to all classes, but I can probably safely assume they do. I also didn't test how damage is calculated for hired mercenaries, but from the test we can also probably assume it is on hit.
An easy way to test this yourself would be to go into Act 1 of any difficulty (I suggest the highest you can go). Then go outside the first area with INT items unequiped and use abilities on the zombies in that area and then equip them mid fight, and hitting enemies with abilities like Soul Harvest and looking at the damage increase. You can display your damage dealt to enemy creatures by going to Options -> Gameplay -> and Checking "Display Damage Numbers"
Best Answer
Yes, DoT spells can crit in Diablo 3.
Here is a link to some research, that shows the following facts:
There is a list of spells tested in the thread: http://us.battle.net/d3/en/forum/topic/5911881247