The first question: Only close combat attacks.
The upgraded Rune of Fire, "grants... a Strength 4 Breath Weapon", so, being a separate weapon, would be unaffected by other weapon runes.
The second question: All attacks.
The Daemon Slayer rule states that, "Any ward saves successfully made against Attacks by the Slayer must be re-rolled.
The Dragon Slayer rules similarly states that, "When attacking monsters, the Slayer's attacks have the Multiple Wounds (D3) special rule.
The rules apply to Attacks, so would work for shooting (you never know...), CC, Breath Weapon (see above), belching or barstools.
The third, well, can't really help you there. The combined rule states, "wizard", so it wouldn't affect a bound spell (even one cast by a wizard, see p.37 of the rulebook) under RAW, as the bound spell effectively casts itself. The rune can be used to dispel a bound spell, there's nothing in the first part preventing it. There's no FAQ as far as I can see, so all I can say is start harassing GW for an answer and if your opponent takes advantage, just record the Grudge and avenge accordingly. With a barstool.
I have no Gazelle.