Осторожность нужна и дома: не оставляй нож на краю стола, чтобы ребёнок не порезался.

Russian often allows flexible word order. Here Осторожность is placed first to emphasize the topic (Caution / carefulness).
Нужна is the short-form adjective нужный (“needed/necessary”) and it agrees with осторожность in:

• gender: feminine
• number: singular

So: осторожность (fem. sg.) нужна (fem. sg.).

Осторожность is in the nominative. This is a common impersonal-like pattern in Russian:

• X нужен/нужна/нужно/нужны = “X is needed/necessary”

If you want to add “for someone,” you can add a dative:

• Осторожность нужна (кому?) всем. = “Caution is necessary for everyone.”

But the core construction doesn’t require a dative.

Here и дома most naturally means “even at home / also at home”: caution is needed not only elsewhere, but at home too.

Russian и can mean:

• plain “and” (joining items)
• “also/too”
• “even” (with this kind of emphasis)

Context decides; here it’s the “also/even” meaning.

The colon introduces an explanation or an example of what “caution at home” means. It’s like English:
“Caution is needed at home too: don’t leave a knife…”

You could also see a dash in some styles, but the colon is very standard for “here’s what I mean.”

Both are possible, but they differ in tone:

• не оставляй (imperfective imperative) = a general rule / don’t do this habitually / don’t be doing this
• не оставь (perfective imperative) = don’t do it (as a single completed action), often more “pointed” or situational

In safety advice and instructions, the imperfective imperative (не оставляй) is very common.

не оставляй is the ты (informal singular) imperative.
For вы (polite singular or plural), you’d use:

• не оставляйте нож на краю стола…

Because it’s the direct object of оставлять (“to leave”). For inanimate masculine nouns like нож, the accusative singular equals the nominative singular:

• nominative: нож
• accusative: нож (same form)

So you identify it by function in the sentence, not by a different ending.

Both can exist depending on the noun and its declension patterns. край often has the prepositional singular на краю in this meaning (“on the edge/brink”).
на краю стола is the standard, natural phrase for “on the edge of the table.”

Because на краю requires “of what?” to specify which edge:

• край (чего?) стола = “the edge (of) the table”

So стол → стола (genitive singular).

чтобы introduces a purpose or prevention clause. In this sentence it’s basically “so that / so … doesn’t happen”:

• не оставляй …, чтобы ребёнок не порезался
  = “don’t leave … so that the child doesn’t get cut”

It often pairs with a negative in safety warnings: “do X so that Y doesn’t happen.”

Because ребёнок is the subject of the verb in the чтобы clause:

• ребёнок не порезался = “the child didn’t get cut”

So it’s nominative (subject case), even though the whole clause depends on чтобы.

In Russian, after чтобы, it’s very common to use a past-tense form to express a desired/undesired result (especially with perfective verbs), even when the meaning is future.

So чтобы ребёнок не порезался means “so that the child won’t end up getting cut.”

A more “future-looking” alternative also exists:

• чтобы ребёнок не порезался (very common, result-focused)
• sometimes you may see чтобы ребёнок не порезался preferred over forms like не порежется in this structure

порезаться is a reflexive verb meaning “to cut oneself / to get cut.” The -ся marks that the action affects the subject:

• порезать (someone/something) = “to cut”
• порезаться = “to cut oneself / to get cut (accidentally)”

So ребёнок не порезался = “the child didn’t cut himself / didn’t get cut.”