1) What is a multicellular organism?  

Obviously, one cell such as paramecium, happily swimming around is a single-celled
 (unicellular) organism.  But, what about a colony of yeast cells?
The answer is that just because cells are attached to each other, it doesn't mean that the
 organism is multicellular.  In a multicellular organism, there is usually differentiation of
 cells, and a single cell, would not be able to survive on its own.  In the case of yeast,
 with enough food, a single cell is a perfectly stable organism, independent of the colony.

2) For autotrophs, does the nucleus also contain almost all of the DNA?

The distinction that the professor was alluding to was that in autotrophs, the nucleus is
 not necessarily the largest organelle.  In plants, the central vacuole may be bigger than the nucleus.  
Why "almost all of the DNA"?  Because the mitochondria and in plants mitochondria and
 chloroplasts contain some DNA.

3) Do certain species of bacteria only move a certain way (i.e. only counter-clockwise or only clockwise)?

Perhaps you are asking about the clockwise versus counter-clockwise motion of flagella? 
 As far as I know, bacteria and protists,  either move forward, backwards, or tumble.  It is
 the flagella which may move in a clockwise fashion, propelling the cell forward.  Often,
 a cell will reverse the motion of flagellum, and tumble, thus changing direction of the
 movement.

4) In the ER when proteins are made, where/when are the proteins tagged 
with sugar? is it in the ER or in the Golgi?

Proteins have the carbohydrates added to them in the ER, but the sugar groups of these
 glycoproteins will often be modified.

5) Smooth ER is the site for lipid synthesis.  What organelle in the SER is responsible for this?

Smooth Endoplasmic Reticulum is itself an organelle.  SER is the organelle responsible
 for lipid synthesis, using specialized proteins that are located in the SER.

6) Is it correct to say that the ribosomes Rough ER synthesizes proteins which are to be
 secreted out of the cell, and the free ribosomes synthesize those to be used in the cell?

No.  While it is true that proteins which are destined for secretion are synthesized by the
 RER-bound ribosomes, certainly not all RER proteins are secreted.  Some will end up in
 the lysosome, some will be retained in the RER itself, and so on.

7) What is taking place at the lumen at the ER?

Lumen simply means inside.  So everything that happens inside the smooth ER (lipid
 synthesis, carbohydrate metabolism, detoxification) and the rough ER (protein folding,
 glycoprotein formation) constitutes the activity of the luminar space of the ER.

8) free ribosomes make proteins that will not go through the golgi complex?

That is correct.  Cytosolic ribosomes manufacture proteins that are likely to remain in the
 cytosol, or go to the nucleus/mitochondria/peroxisome, but NOT to the golgi.

9) how DNA in human's mitochondrion is passed on to the next generation?  Are they in the human egg only? If that is the case, children only have the maternal mitochondrion's
 DNA?

Indeed, human mitochondrial DNA is maternally inherited.  Although there are
 mitochondria in sperm, the genetic information is not passed down to the offspring.  This
 is precisely what makes mitochondrial DNA so valuable for population
 genetic/anthropological studies.

10) Which one lysosome or peroxisome degrade "old" protein, RNA, DNA ?

Lysosome has the DNA, RNA, and protein cleaving enzymes, while peroxisome
 functions mainly to oxidize toxic molecules.