After I do the quest with the BOS and start the conspiracy for getting the current elder removed, no speech dialogues show up.When do they appear again to do more missions?

You have to complete "still in the dark" and then do another quest to actually become a Brotherhood member.

You could also complete Arcade Gannon's companion quest and get PA training that way, but that won't happen until near the end of the game.

Wait a few days for Hardin to properly take over, then speak to him about joining. Arcade's quest also earns you training.

Hardin dosent need to be leader to get power armor training....
the virus quest foiled me every time so I skipped firing the old elder...seems you get better outcomes later on if you leave him in place.