The feat War Caster allows you to cast a spell when an enemy provokes an Opportunity Attack from you. The spell must target only the provoking creature, and can only have a casting time of 1 Action.

For the sake of abilities like Sentinel (which causes Opportunity Attacks to reduce a creature's speed to 0), does my spell cast by War Caster still count if it uses an attack?

No, it's not an opportunity attack

The relevant phrasing in the Sentinel feat (PHB, p. 169-170) is:

When you hit a creature with an opportunity attack, the creature's speed becomes 0 for the rest of the turn.

So the prerequisite for this feature is that you must hit a creature with an opportunity attack.

The relevant phrasing in the War Caster feat (PHB, p. 170) is:

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

While the trigger is a situation that provides an opportunity attack, it specifically states that no opportunity attack is made, and you instead us a reaction for the spell casting. Thus, the Sentinel feat is not triggered.

No it is not.

The details explicitly say 'rather than making an opportunity attack'. However, either an opportunity attack or the players spell would use up the one reaction per turn.

PHB p170

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack.