It needs to be repotted in about a 3 gallon pot or bigger. Are the leaves sharp? It's hard to look at it side ways. Can you straighten the picture? It looks like a Yucca or a Dracena.

Here is a close up of the leaves.

The bark might be a better thing to look at. I'm going with Yucca.

Yucca gigantea (syn. Yucca elephantipes, Yucca guatemalensis).

Now my poor Yucca is in rough shape. Many of the lower leaves are wilted and yellowing. Some leaves have brown at the tips. What have I done to this poor plant? 2 weeks ago it was in a moving truck for an hour in -6c temperatures. What should I be doing to keep it healthy?
Any help would be greatly appreciated.

That's almost certainly cold damage. Keep it warm and well lit. Water only when dry an inch down into the root ball.

Should the yellowed leaves be removed? Do you think it will survive?

Should make it, but some tip dieback may occur. Yellow leaves can stay or go your choice.