Do stalactites and stalagmites ever run into each other? If they do does it just form some kind of pole?

Yes. When the stalactite and stalagmite connect it is called a column.

"The stone formations which grow up from the cavern floor are created in the same manner. Sometimes large droplets of water filtering through the cave roof don't have time to evaporate before they roll down a stalactite and drip off to the cave floor below. That's why stalagmites usually form directly below stalactites and continue to grow as more droplets fall from above. In some cases (such as the grand Pipe Organ formation at Howe Caverns), stalactites and stalagmites will actually grow together to form columns."

When a stalactite reaches a stalagmite, it then forms a column or pillar. Those are the technical terms.

Yep. They'll join if left alone, and the mineral-laced water keeps dripping.

Yes, they sure do! Take a look at some pictures at this website ==>http://www.earthscienceworld.org/images/search/res...

Sure they do - they form columns.