What is an attempted swiftstrike based on? Does it just automatically check for one EVERY parry? What factors/stats can affect it and cause it to trigger more/less? Weapon type? Weight? Str/dex? Or is it strictly skill based and checked for on every parry? Does haste/slow have any effect?

Like riposte, it's checked for on every successful parry. In a sense, the biggest factors to swiftstrike are often therefore whatever helps you parry, since if you don't parry the attack you have a 0% chance to swiftstrike.

You also have to be able to see your opponent (generally) to swiftstrike. (Note, for example, the Sightless Eye edge which allows you to semi-circumvent this restriction.)

Once you clear those hurdles, non-marginal factors are:
Swiftstrike skill
DEX
Haste/slow exerts a very small effect (directly, anyway -- it's really overshadowed by, say, how much haste helps you parry and therefore swiftstrike more)
A couple other generic fighting skills (basically, SS works a little better for a warrior, who has stuff like third attack, than it does for an invoker that doesn't)

For a sword spec is it possible for both swiftstrike and riposte to trigger on the same parried attack, or is it one or the other or neither?

You can't succeed at both on the same attack. If whichever one is checked first succeeds that parry is finished.