Question about Hypogean Gaol

No.

There are a few triggers in the previous stages of the quest, but being on the last one you should be safe. And be careful not to kill her by mistake ; she can be killed by mobs, or if you use the Choir Bell on her.

The first run. I would strongly discourage continuing to NG+, because then the DLC areas will get harder just like everything else.

You have access to the DLC until you initiate the fight with the final boss. I would recommend occasionally checking in on the DLC and to see when it stops feeling too difficult for you, at which point you can do some or all of it.

The following spoiler about when the endgame begins is worded in the least spoilery way I can manage.

There will a substantial cosmetic change to the Hunter's Dream when the final boss becomes available, but you are not cut off from the DLC until you speak with an NPC there.

The one after this line is very specific.

After defeating Mergo's Wet Nurse in the Nightmare of Mensis, the workshop in the Hunter's Dream will catch on fire but remain usable. You will be able to speak with Gehrman in a previously inaccessible area. The dialogue choices you're presented with initiate a boss fight with him, or a cutscene where you let him kill you. The latter immediately boots you out to NG+.