[RPG] Does a failed Stabilize action count as a failed death saving throw

No,

Assuming the power used was Summon Thought Servant ^DDI, it works exactly like a Heal check.

Opportunity Action: Triggered when a dying ally adjacent to the servant starts its turn; targets the triggering ally; the servant makes a Heal check to stabilize the target or to allow the target to use his or her second wind

In a Heal check, the subject is told to stop making death saving throws. There is no option to willingly make them. In addition, the subject is still likely unconscious and cannot take any actions.

Stabilize the Dying (DC 15): If the subject is dying, the subject stops making death saving throws until it takes damage. Being stabilized does not change the subject’s current hit point total.

Note that there is no stabilized condition and being stabilized does not remove the Dying condition. The character is no longer required to make Death Saving Throws.

This means that a stabilized character is still a valid target for effects like Energizing Strike ^DDI and Get Up! ^DDI

The Dying condition is removed from the character only when the character regains hit points. Being stabilized is not enough.

The easiest way to gain advantage on a death saving throw is to use inspiration.

If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. Spending your inspiration gives you advantage on that roll.

Alternatively, you could be a Wild Magic Sorcerer, and use Tides of Chaos. Well, probably.

Starting at 1st level, you can manipulate the forces of chance and chaos to gain advantage on one attack roll, ability check, or saving throw.

The 3rd- level spell Beacon of Hope explicitly grants advantage on death saving throws for its duration.

Fighters can use Indomitable to reroll a failed saving throw - not precisely the same as advantage, but pretty close.

Beginning at 9th level, you can reroll a saving throw that you fail.

There are almost certainly more options that I've missed - getting advantage on things is a pretty fundamental part of 5e.