Mi primo asegura que el pasillo será más seguro después de decorar las paredes.

It literally means "My cousin assures (that)". Here asegurar doesn’t mean “to ensure” (garantizar) but rather “to state/claim/assert” something. A natural translation would be:

• “My cousin claims that…”
• “My cousin says that…”
• “My cousin assures (me) that…”

Because asegurar used with a subordinate clause requires the conjunction que to introduce what is being asserted. The pattern is:

• asegura que + [clause]
  Just like English “claims that,” “says that,” etc. Without que, the sentence would be incomplete.

Será is the simple future of ser (“to be”). The speaker isn’t describing a current fact but predicting a future condition:

• es más seguro = “it is safer” (now)
• será más seguro = “it will be safer” (later, after the walls are decorated)

Yes. Va a ser más seguro uses the periphrastic future and is very common in spoken Spanish, expressing a planned or likely outcome based on current information. Será más seguro (simple future) often sounds a bit more formal or expresses a firmer prediction. Both mean “it will be safer.”

Because después de is a preposition, and prepositions must be followed by a noun or an infinitive, never a finite verb. Therefore you say:

• después de decorar las paredes (“after decorating the walls”)

If you want a full clause with a conjugated verb you must add que + subjunctive:

• después de que decoremos las paredes

Absolutely. You can attach the object pronoun to the infinitive:

• después de decorarlas

Or with al (“upon”):

• al decorarlas

Both replace las paredes with las.