Can i discharge an touch spell with an unarmed strike?
If you are holding the charge, yes.
Holding the Charge: "If you don't discharge the spell in the round when you cast the spell, you can hold the charge indefinitely. You can continue to make touch attacks round after round. .... Alternatively, you may make a normal unarmed attack (or an attack with a natural weapon) while holding a charge. In this case, you aren't considered armed and you provoke attacks of opportunity as normal for the attack. If your unarmed attack or natural weapon attack normally doesn't provoke attacks of opportunity, neither does this attack. If the attack hits, you deal normal damage for your unarmed attack or natural weapon and the spell discharges. If the attack misses, you are still holding the charge."
Can a creature with drain ability drain with a touch attack, rather than with slam?
Ability Damage and Drain (Ex or Su): "Some attacks or special abilities cause ability damage or drain, reducing the designated ability score by the listed amount. Ability damage can be healed naturally. Ability drain is permanent and can only be restored through magic. Format: 1d4 Str drain; Location: Special Attacks or individual attacks."
If it specifies slam, then it's just slam.
An ability drain/energy drain creature with the Spellstrike Magus Class Feature can use its slam attack with a spell?
If the creature is a magus, and it casts a spell from the magus spell list with a range of "touch", then it can use Spellstrike to deliver that spell with it's weapon, in addition to all the normal options available.
A supernatural or extraordinary ability is not casting a spell, so it doesn't work with Spellstrike.
However, you don't need spellstrike to deliver a touch spell with a natural attack (or unarmed strike). So if the creature has a held charge, it can slam someone and deliver the spell.
Here's A Guide to Touch Spells, Spellstrike, and Spell Combat.