The only role that can viably buy an early Vampiric Scepter is a jungler. This is because lifesteal procs off auto attacks, and in early game you should not be auto attacking creeps because it will push your lane too much.
One of the main ideas behind stacking Doran's is that it provides a cheap advantage for early game harrasment. If two champions with similar base stats are harassing each other, the champ with 2 Doran's will have greater sustain in lane and most likely will win the harrasments. Having more health allows you to win the skirmishes against your opponent, survive jungler ganks, last hit easier, and stay in the lane after a failed jungle gank or harassment.
This does put you behind for getting your first big purchase item, however given that a B. F. Sword (the most common big purchase for AD champs) is 1550g, spending an early 475g on a second Doran's, and consequently getting greater CS (note that after ~25 CS the dorans has paid for itself) and champ kills, can actually allow you to get that B. F. Sword sooner.
This is a pretty broad question but I'll tackle a couple ways ignite could be used best. First Ignite has 2 effects, many people forget about the second(go read), and each of these effects have best uses.
The damage over time part (50 + (20 * lvl) true damage):
This is good for ending someones life when they are low hp and may get out of range soon. This is bad when that someone has a shield or some other mechanism of avoiding damage while low hp (its true damage but you can still avoid it). Ex: Shen's Feint, Janna's Eye of Storm, barrier, etc.
This is good when you need/want a bit of extra damage going into a 1v1 trade (you don't always need to use it to finish people off). This is bad if you use it too early because then they will either just run away(leaving you without an ignite).
The damage received by Ignite is true damage, meaning it ignores armor/magic resists. This is good for a bit of extra damage on an enemy with armor/magic resist. It is bad when that enemy is a tank with a lot of hp because you may misjudge the actual life they have left, and Ignite may be better used on a squishier target.
Before moving onto the second portion you mentioned Tryndamere being a good choice to use ignite on, this is true, but not just because he's at low hp when Endless Rage ends and it will tick to kill him. It is due to the second effect, the heal reduction, reducing the amount he can heal from his Bloodlust. This leads to the second portion of the spell.
Reduces health regen and healing effects received by 50%:
This is good for when the other team has a healer or you're fighting someone with a self heal. Often times using ignite early on an adc against a team with Sona, Soraka, Nami, etc will help to win fights because the target will receive less heal. However, if the other team has no direct heals this portion of the ability will only affect Lifesteal,regen, and Spell vamp. In team fights as mentioned above this is useful to use on priority targets earlier.
General Notes
Cleanse removes Ignite's damage, but the heal reduction remains. Keep this in mind when using it on a target with cleanse, as you'll only get half the benefits of this spell.
Ignite does not stack with other ignite's(the duration will just get refreshed), if you have a lot of ignites on your team already, it may be worth looking into other summoner spells.
Ignite does not play well with other heal reduction effects, meaning it doesn't stack with them. Miss Fortune's Impure Shots, Tristana's Explosive Shot, etc.
Fun fact: Tryndamere's full fury Bloodlust heal is 300 + 150% AP, Ignite's damage over time at max lvl is 410, so just remember without the heal reduction portion on Ignite, Tryn would most likely be able to survive.
Best Answer
My personal opinion is that you can't and you actually don't really need to counter ignite.
I mean you could consider cleanse to get rid of the healing debuff so when you get ignited you can just drink a potion and get the full healing it offers. But unless you're facing a CC heavy team having cleanse in mid is just not worth it. It's better if you get an ignite as well (and a flash ofc).
What you wanna do is be careful not to get yourself in a position to die from an ignite. Cause if you do there's not really much you can do. Especially early game. Not many items you can buy to counter it since its early. Like you said healing spells don't do much.
Damage absorbing shields should do you more good there. Like barrier (not worth it imo, much like cleanse) or skill shields like lux's W etc.
All in all ignite is a strong spell and early game the only thing that can save you is awareness of danger and backing out soon enough.