"It is stuck firm."

I think that youre over analyzing it . A native speaker would often interchange the two phrases, "Stuck firm" or "stuck firmly" whether or not one was more "accepted" than the other . Authors like Mark Twain and ELmore Leonard have used more vulgar english in their work merely for effect . However, Out of context its hard to see whether thats the case in your clip.

"Stuck fast" is imo a more common phrase than "stuck firm".

"Firmly stuck" sound okay, though.

So it's probably an adverb. It's modifying the verb.

Now I'm not sure about this:

It- subject
is- verb
stuck- verb, past participle
fast- adverb.

Yes, that's it.

I believe that 'fast' may truthfully be used as an adjective when describing something jammed or stuck. It is the opposite of loose. Believe me. I went out with a girl from Yorkshire and they still use the word apart from the stock phrase "stuck fast". How firmly is it jammed? It's fast. See "playing fast and loose".

A door can be "shut fast" = shut tight, tight shut.

And a fortress can be referred to as "a fastness".

"The bandits retreated to their mountain fastness". A secure place.

If you tie something up you make it fast. You can make a person or thing fast to a tree for example.

Very good. Or a boat to a jetty.

And you "fasten" things.

So, if fastener is the comparative, would fastenest be the superlative?

To the Fastnet?

And you can stand fast either literally or metaphorically.

Stand Fast? I don't mind if I do.

Fastener is a noun. Faster is the comparative.

Combat by Romain Rolland (1935)

They get stuck faster with every movement of their wings." Annette saw this in the case of her bumble-bee, her Timon. In vain did he buzz and spread fear around him. He could not escape. And he knew it! Annette was witness of his furies.

A painter's camp in the Highlands, and Thoughts about art by Philip Gilbert Hamerton - (1862)

I tried it once with cold water, but it would not do, the fat stuck faster than ever to the plates. I know better now, and heat the water, which melts the fat, and so I get my plates tolerably clean. How I do admire and respect all scullery-maids!

Lord Byron (Don Juan)

However this may be, 't is pretty sure The Russian officer for life was lamed, For the Turk's teeth stuck faster than a skewer, And left him 'midst the invalid and maimed (etc)

...and fastest the superlative

John Wesley by AC Outer (1980)

Methodist was the label that stuck fastest. Wesley disliked it, but with characteristic aplomb, he accepted it as a badge of honor

Cry Macho: A Novel by N Richard Nash (1975)

...the thing that had stuck fastest with the kid was that wetbacks occasionally got killed. Something had happened to the boy. He had lost some of his childhood recklessness, his fool's courage. He was suddenly unsure of himself

It was a joke, Bubba . . .

Forgive me: it seemed quite plausible that you were serious.

Will ye take a wee dram, Doctor Finlay?

Drive Safe: In Praise of Flat Adverbs

http://www.merriam-webster.com/video/0030-flatadverbs.htm?&t=1361474223

There's nothing to forgive. But note that i said "fastener" and "fastenest," not faster and fastest.

I saw that you did, and under the impression that you were serious, I proceeded to give the correct comparative and superlative.

By the way, when i asked you to forgive me, I was being facetious.