Saya memegang dinding saat naik tangga karena tangga itu licin.

Memegang is the active verb form with the prefix meN-, which is the neutral, standard way to say to hold in a full sentence.
Pegang is the base form and is common in:

• commands: Pegang dinding! (Hold the wall!)
• very casual speech where prefixes get dropped: Saya pegang dinding... (colloquial)

meN- typically marks an active transitive verb (doing an action to an object).
Here, the verb takes an object: memegang dinding (hold the wall).
It also affects spelling: meN- + pegang → memegang (the p drops).

Indonesian doesn’t use articles like a or the. Dinding can mean a wall or the wall, and context supplies the specificity. If you want to be more specific, you can add something like:

• dinding itu (that wall)
• dindingnya (the wall / his-her-its wall)

Both mean when / while, and both work here. Common differences:

• saat feels a bit more like at the moment of / while and is very common in writing and formal-ish speech.
• ketika is also common and can feel slightly more “narrative” or explicit.

Your sentence would still be natural with ketika: ... ketika naik tangga ...

It’s an elliptical (shortened) clause. Indonesian often omits the subject when it’s obvious from context.
Expanded, it’s basically: saat (saya) naik tangga = when (I) was going up the stairs.

It can mean go up stairs or more generally go up a ladder/steps, depending on context.
If you want to be unambiguous:

• naik tangga usually reads as go up the stairs in everyday contexts.
• naik tangga (lipat) could be climb a ladder if specified.

Both are possible, but they differ in nuance:

• naik tangga is the most common, natural way to say go up the stairs.
• menaiki tangga (with meN- and -i) is more formal/explicit and emphasizes the action directed onto the stairs (roughly ascend the stairs).

In daily speech, naik tangga is preferred.

Karena means because and introduces the reason clause. It can appear:

• in the middle (as in your sentence): ..., karena ...
• at the beginning: Karena tangga itu licin, saya memegang dinding saat naik tangga.
  Both are correct; starting with karena often feels a bit more formal/written.

Itu here works like that/the to point to a specific staircase already known in context.
So tangga itu ≈ the stairs (in question) / those stairs.
Without itu, tangga licin could sound more general: stairs are slippery or the stairs (unspecified) are slippery.

It can be either, but with tangga most learners should read it as the stairs unless context suggests a ladder.
Also, Indonesian doesn’t mark singular/plural strongly here—tangga itu can refer to one staircase (a set of stairs).

Indonesian often doesn’t use a copula like is/are. Adjectives can function as predicates directly:

• Tangga itu licin. = The stairs are slippery. You can add emphasis with sangat / sekali:
• Tangga itu sangat licin.
• Tangga itu licin sekali.

You can, but it’s easier to understand if you keep saat naik tangga close to the action it modifies.
Best options:

• Saya memegang dinding saat naik tangga karena tangga itu licin. (original; very clear)
• Karena tangga itu licin, saya memegang dinding saat naik tangga. (also clear)

Putting saat naik tangga at the very end can make it momentarily ambiguous about what it modifies.