I always wondered what that is. I found one like it and sold it. I believe it had a spring in it. I also found just the top piece. Mine is full of wax. Maybe different. Very strange. Some type of candle. I will take pic.
The first lamp may be a burner for a chafing dish.
The second candle stick is an item as it burns down it will push up due to the spring inside the thing.
Take a look on the end of the knob that turns the wick up. Many times there is manf. info on the end of the knob and sometimes there is a patent date. I dug one from 1906 and this is where I found the info for the date!!!
It might be an old Dietz lamp. They make them still in Syracuse NY. They have been around for a long time and I think they have a museum of lamps on their website. The only thing is....I think they stamp Dietz on the bottom. Just a thought.