Taliban (across the Panj river) has been a threat until recently. But for now they seem to be friends with Russia and Tajikistan has always been a Russian client state.
As of last year but probably since the Taliban came to power, Tajikistan has has military checkpoints along the border. Also, I don't know about for all seasons but during the summer, the river seemed too fast to be crossable along most parts of the border.
I don't think they have. It also wouldn't really make sense for them to do that. The Taliban mostly have issues with Iran and Pakistan so you will see conflicts with these nations most often in the news. I've never heard of them in conflict with any of the other stans. Badakhshan is also quite distant from the core of Taliban power which is mostly in the South. Northern Afghanistan is traditionally Northern Alliance territory.