Dying
DDI is a condition, part of which states that you are unconscious. Unconscious
DDI is a condition as well, part of which says that you can take no actions. Therefore you cannot activate powers unless they explicitly state that no action is needed.
The Death Saving throw is still a saving throw, you only fail if you roll under a 10. Rolling a 20+ is a special exception that allows you to spend a healing surge. Heroic effort can only be used on a failed attack or a failed saving throw. Therefore it could not be used in this situation. (I'd have no problem with a DM allowing it though, it sounds like a cool use to me). Note that it could be used on a roll of 9 or less (a failed saving throw)
It sounds like you have a very acceptable fire-focused intention. I suspect one of the things that is complicating matters is the fact that:
The Mage Wizard can select 2 Encounter powers per level, so that offers some variety. It helps that the DM lets us waive the restrictions on the number of powers we can use per day. I think that's why he makes the monsters so strong in each encounter, so they don't get wiped out easily.
Therefore, much of the normal optimisation advise, which assumes that you're holding to the normal rules starts melting away as the vicious circle of buff and counterbuff begins. (I faced this problem in a "by the rules" game when the DM reacted to the party's increasing optimisation by ramping up monsters, which caused us to optimise more, which...)
From a pragmatic perspective, save ends effects suck. While much of the game is well modelled, there is precious little balance to save ends effects, and the pendulum swings back and forth: standard monsters have little to no defense, but elites and solos become effectively immune as the game design progressed through the monster manuals. It takes a very deft touch in monster creation (if you're creating monsters "from scratch" to respect player agency in the inflicting of status while simply not going "nope!" to either them automatically winning or to them automatically being ignored.)
I, personally, have always enjoyed the more controlly-type controllers, and so my wizards, druids, psions, and invokers have focused on debuffing and forced movement. So long as you rely on effects that are more difficult to shed (either being end of next turn or encounter long) then you can focus on being to reliably land them, rather than inflicting sufficient debuffs to the monster's saving throws (that'll only be countered by the next monster) to maintain the debuff. The same thing is true in the other direction. I've played paladins who granted +9 to saving throws by smiling. This led to the DM completely foregoing the use of save-ends effects until the DM and I agreed to voluntarily limit that feat to a +5 bonus.
My recommendations are:
Nothing is as powerful alone compared to a party that is designed to work together.
Stop focusing on solo optmisation. It's a trap. Instead, try to make sure the party is designed to work together to achieve your desired requirements. Everyone will have more fun, and you're unlikely to bear the brunt of your DM's nerfing alone.
Have a side conversation with your DM: Explore what debuffs he's comfortable with.
Boundary setting is important. If you have a chat over coffee as to what he considers reasonable, you won't find the powers nerfed in the middle of a game. Set up, describe, and agree upon expectations for your character's capabilities such that he knows what to expect (such as to provide you maximum Fun) with the minimum of unpleasant surprises. As 4e is very much combat-as-sport, the joy is in the execution of plans within a chosen narrative (yes, story matters, to provide a need and justification for mechanics) than it is finding unusual solutions to the DM's prepared set-piece battles (many other systems are far far better at simulation).
It's very hard to alter characters in midstream without a retcon. Be honest and do a proper retcon, don't just knudge.
A character is the combination of her parts and their interactions, not just the parts alone. If you're changing a character's rasion d'etre, be honest about it, and change the character completely to fit your new requirements.
Best Answer
Yes, but it's unlikely
It's entirely within the rules to continue attacking a character while he's unconscious, and a GM/monster might decide to do so for a number of reasons. However, there are usually more reasons he won't, both in-game as a monster decision and as a meta-game GM decision.
First, there's a rule
Though more what you'd call a "guideline" than an actual rule.
So GMs are clearly discouraged from kicking people while they're down... but why?
It's a poor tactical choice, or at least makes sense in context
When a PC is already down, the monster is going to be much more concerned about the people who are still able to deal him damage and debuff him.
If the monster has some RP reason to particularly hate the downed PC (nemesis type villain, maybe?) then it could be better justified as a tactical choice—but it's still not a good one.
It's a vicious—and frankly boring—GM choice
You've spent time and effort on your character and unless you're in a group that has decided ahead of time to be excessively lethal, a GM using coup de grace should generally have a good reason it won't make you hopping mad as a player to get taken out so undramatically. It'd be much more interesting for the PC to get dragged off to the monster's lair / laboratory / tropical resort, setting up scenes of dramatic rescue or escape (this is actually suggested in the paragraph following the above DMG quote).
But sometimes it's a good tactical choice, or at least makes sense in context
For example, the Fell Taint monsters heal to full if they successfully kill a creature with a coup de grace. They trade this for dropping their insanely strong defenses (insubstantiality) and being unable to move for some time, so it's tactically interesting.
And not all monsters make good tactical decisions, of course.
In these cases the GM should consider the "vicious GM choice" aspect when deciding if the zombie is going to stop to chow on some brains in the middle of combat, or the evil mastermind takes a moment to skewer an unconscious foe.
And sometimes it's a good and interesting GM choice
PC death can be good for story and character growth, provided the GM and the players trust each other to make decisions the group finds palatable.
Addendum: "Kill him before he heals again" isn't a no-brainer in 4e
In 4e, NPCs and monsters are mechanically distinct from PCs. One of the results is that no NPC has more than one healing surge per tier, and very few have any more healing powers than Second Wind.
This means that unless a monster has prior knowledge of the PCs it has absolutely no in-game reason to expect they'll be able to heal dire wounds repeatedly, or that a particular character will have the ability to grant multiple massive heals to others. Within the 4e ruleset, the instant mid-combat healing of grave injuries is not common: it is the purview of player characters alone. Not even gods have access to the breadth and depth of healing-per-day that level 1 PCs do. This is part of what makes them heroes: their access to unmatched ability and potential.
Of course, a villain that has fought the PCs before, or who has observed or researched them, can reasonably know this. A monster that sees the PCs go up and down like yo-yos may get tired of it and adjust his tactics toward the end of the battle.
Coup de grace?
4e has the coup de grace action, which allows any successful standard-action attack power that targets a helpless creature to be an automatic critical. Regardless of whether the damage dealt should kill the target, the target dies if the damage is at least its bloodied value.
There is no good reason for a monster attacking a helpless creature with lethal intent not to use this action, aside from being blindingly stupid or some similar handicap. However, a GM might choose to 'forget' this rule if he wants to attack a helpless player to indicate the seriousness of the monster's threat without practically guaranteeing a PC kill.