Translation:Please come up to the 15th floor.
I think you need "up" because of the 上 particle, which is one of the concepts they are trying to teach in this unit.
Yes, 上 (rise, ascend, go up) is actually the verb in this sentence. 来 is a directional complement after the verb showing that the motion is toward the speaker.
No, that's not true. This is not about grammar as much as it is about the way of using the language, 十五楼 means the 15th floor, no need for 第.
I do not believe it is correct to use 第 in this way. 第 is for events that occur in sequence (my first time going to China, my second time going to China). Things that happen to be physically laid out in an ordinal sequence (such as Building One, Building Two) as well as days of the month (the tenth, the eventh) use 号. The measure word for layers or levels of something (floors of a building, layers of a cake) is 层, but I am not sure whether 层 is actually needed in this sentence.
Could this also mean "Come up 15 floors," if someone is on the 5th floor and needs to come up 15 floors to the 20th floor?
No, 15樓 is specificly saying the 15th floor. Come up 15 floors 往上爬15層, here use 層 instead of 樓