[RPG] Is it safe to bring an item like a Bag of Holding into a Genie Warlock’s Bottle

Yes, a portable hole is a "similar item," and when put into another portable hole will destroy both and create a planar rupture.

It's the "nesting" of extradimensional space that causes the havoc, not any properties particular to these magic items individually or as a set.

Note however that there is something particular to the kind of extradimensional space attached to these items: the nesting problem only manifests relative to extradimensional space created by such magic items. They'll all play havoc with each other, but you don't have to be paranoid simply carrying such items into other planes, pocket dimensions (natural or spell-created), and the like. Whatever space these items connect to, it's not the kind of space you can wander into unknowingly while planeswalking.

No the bag will not burst, because Rope Trick is not an item

The key word in the caveat you cite is "item":

Placing a bag of holding inside an extradimensional space created by a Handy Haversack, Portable Hole, or similar item instantly destroys both items and opens a gate to the Astral Plane.

The two examples the text gives are also items.

However, the extradimensional space created by Rope Trick is created by a spell, not an item, and so does not fall under that clause.

There are many spells that generate extradimensional spaces (such as Mordenkainen's Magnificient Mansion)--if the designers intended the Bag of Holding to explode when it entered such spaces, they could have easily made the text refer to any extradimensional space, or used spells as examples.