Spells like Moonbeam, Flaming Sphere, and Call Lightning allows you to use either an Action or a Bonus Action (depending on the spell) to trigger a specific effect on subsequent turns. Does triggering these effects count as casting a spell?
I'm fairly sure that one can't Counterspell them once it has been casted, but would it trigger the Mage Slayer feat, for example? (Assuming all conditions for Mage Slayer are met)
From the section 'Actions in Combat':
When you take your action on your turn, you can take one of the actions presented here, an action you gained from your class or a special feature, or an action that you improvise. [...]
Cast a Spell
Spellcasters such as wizards and clerics, as well as many monsters, have access to spells and can use them to great effect in combat. Each spell has a casting time, which specifies whether the caster must use an action, a reaction, minutes, or even hours to cast the spell. Casting a spell is, therefore, not necessarily an action. Most spells do have a casting time of 1 action, so a spellcaster often uses his or her action in combat to cast such a spell.
You are casting a spell only when you take the Cast a Spell action. The actions described in those particular spell descriptions are not the Cast a Spell action, rather they are "an action you gained form a special feature".
Counterspell's trigger says:
which you take when you see a creature within 60 feet of you casting a spell.
Since the actions mentioned in these spell descriptions are not "casting a spell", counterspell cannot be used.
Similar reasoning as counterspell. The first point of Mage Slayer says:
When a creature within 5 feet of you casts a spell
As you've stated, those continuing actions/bonus actions on an active spell are just that - continuing actions. You aren't casting a new spell and/or using components. You are just using a function of the spell.
which you take when you see a creature within 60 feet of you casting a spell
Counterspell is only a viable reaction if a creature is casting a spell. Without that specific action, there is nothing to counterspell. That spell is really just about attempting to stop someone from completing a casting.
If you want to stop a currently active spell, your best bet is dispel magic or something similar that works against currently active spells or magical effects.
