1. Battle of Dale: Things were looking grim for everyone in Middle-earth until Frodo reached Mount Doom and the Ring was destroyed.

Yes, and particularly because there was no hope of rescue by any other army.  With Gondor tied up down south, everyone else was on their own.

3. Under Siege: what do you think it was like for the Dwarves and Men sequestered together in the Lonely Mountain?

Well, they had water, and whatever provisions they had stored against a seige, but it is doubtful whether they had much warning of the attack, so there couldn't have been much provision.

4. Endure: How safe were they? How long could they hold out?

Until the wine/beer ran out.  Or a couple hours later, at the most.

4. Battle of Dale: Any comments on the Battle of Dale in the War of the Ring?

I remember being surprised when I first read the Appx and saw this.  It had never occurred to me that other kingdoms were having big problems, and that Lorien and Erebor were beseiged.  It's almost too much to believe, though, that  they both lost their leadership in the battles.  That means that essentially every kingdom in ME had a big change in  command within days. 

5. Crowning: Who do you think each Kingdom sent to the crowning?

Hard to say.  The text says ambassadors, which would suggest the most important personage they could spare, but for something as important as the ROTK, I would consider sending the new king even if he's busy getting his administration going. 

6. Friendship with Gondor: describe the relationship of Erebor have with Gondor? And Dale’s relationship with Gondor? How deep were these Friendships?

Probably pretty deep.  It can't be a coincidence that all the new kings and leaders had some connection with Aragorn or a member of the Fellowship.   These ties of friendship would likely allow peace to break out all over.

7. Ever after: "realms remained ever after, as long as they lasted, …" At first the term ‘ever after’ implies the standard fairy tale ending of: ‘They lived happily ever after’. Why qualify ‘ever after’ with ‘as long as they lasted’? What is being implied? Is this another application of Tlien’s famous ambiguity?

It's both ambiguous, and a standard fairy-tale ending, unless you count the one about the purse of mouse fur, which to this day I don't understand.  I think there is a wink here of JRRT reminding us that this is stuff of legend.

8. As Long as they Lasted: How long could these realms have lasted?

At least 50 or so years till people started squabbling again.  Since they are gone now, then the answer is between  maybe 50 and 3000 years, which are both the same number in geologic time. :)

Some who wander are lost.