Spikeren sitter fast i gipsveggen.

Because spikeren is the definite form: spiker = a nail, spikeren = the nail. Norwegian often uses the definite form when talking about a specific, already-known object in the situation (e.g., the nail we’re looking at).

Sitte fast is a very common Norwegian expression meaning to be stuck / to be firmly fixed.

• sitte literally means to sit, but it’s also used about things being positioned / attached / located in a stable way.
• fast = fast, firmly, stuck.
  Together: sitter fast = is stuck fast / is firmly stuck.

Yes, Norwegian often chooses the “position verb” based on how something is oriented:

• sitte fast: attached or fixed in/onto something (nails, stickers, lids, things wedged in).
• stå fast: “stands” firmly—often upright or fixed in place (can be used more abstractly too).
• henge fast: hanging and stuck (something caught on a hook/branch).
  For a nail in a wall, sitter fast is the most natural.

Because the nail is in the wall material, not merely on the surface.

• i = in/inside (embedded)
• på = on (resting on the surface)
  A picture would typically be på veggen (on the wall), but a nail goes i veggen (into the wall).

gips = gypsum / plaster and vegg = wall.
A gipsvegg usually refers to a drywall/plasterboard wall (common interior wall type). The compound is written as one word: gipsvegg.

Norwegian commonly forms compound nouns as single words: gips + vegg = gipsvegg. Then the definite ending is added: gipsveggen = the drywall wall.
As a rule: if English uses “X wall / X book / X problem”, Norwegian often makes it one word: Xvegg, Xbok, Xproblem.

vegg is masculine in Bokmål, and the common definite singular ending is -en:

• en vegg → veggen
  You may also see vegga in some dialects or in informal style, but veggen is the standard Bokmål form.

Yes, but it changes the nuance:

• i gipsveggen (definite) = in the (specific) drywall wall
• i en gipsvegg (indefinite) = in a drywall wall (any such wall, not a specific one)

Yes, Norwegian allows that. Main-clause word order follows the V2 rule (the finite verb is in position 2), so you can front the location for emphasis:

• Spikeren sitter fast i gipsveggen. (neutral)
• I gipsveggen sitter spikeren fast. (emphasizes where)
  Both are correct.

It’s present tense, but present tense in Norwegian often describes a current state without focusing on “right now”. Here it mainly means the nail is currently in a stuck/fastened state. It can describe a general situation as well.

You can, and it’s understandable, but it sounds less idiomatic for this physical “stuck/embedded” idea. Norwegian prefers the verb phrase sitte fast for this meaning. Er fast can work, but sitter fast is the natural choice.

A rough guide (varies by dialect):

• spikeren: SPEE-ke-ren (stress on spi-)
• gipsveggen: GIPSS-veg-gen (stress on gips-)
  Also note the double consonants affect the rhythm: -gg- in veggen typically signals a shorter vowel before it.