[@darkandstar] Since you are second guessing yourself, I could look over your rulings. You may ignore my opinions on the matters if you see fit.

Yes, as the guard made his grapple attack earlier on his turn, Tarhun will make their contesting roll as the first thing. Probably with Athletics, giving them a +6 on the roll.

While rolling with disadvantage seems okay at first, my call would be to let it be a single roll. Why? Because the grapple was the guard's action and Tarhun was not occupied during that time. In case of Tarhun grappling them back, well Grapple is an action and you only have one of those per turn.

His damage is grossly off as it is multiplied with the modifier instead of adding it (and the hit was not a critical with a 19, not that that would still imply multiplication). And yes, as it was an improvised weapon, he would only have +4. He has probably simply forgotten this detail, it happens. Of course, if they are intent on grappling my call would be to not let them attack on the same turn, but I am not calling the shots here.

Throwing on the next turn seems fair and correct to me.

Also, the effects of Grapple are as follows (PHB Appendix A, p.290):
[list]
[*] A grappled creature's speed becomes 0 and they cannot benefit from any bonus to their speed.
[*] The condition ends if the grappler is incapacitated.
[*] The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the [i]thunderwave[/i] spell.
[/list]
It does not quite take away the whole turn, as taking away Tarhun's speed would not really affect their plans. Overturning a grapple probably has some rules somewhere in case Tarhun fails the check, but I couldn't find any with a quick look at PHB and DMG, so it's your call. So yeah, you are pretty much correct  with things here, a little more lenient on the action economy side but hey, Rule 0 (The DM is always right) and Rule of Cool apply.