In Janus, what are Andrea's traits, strengths, or weaknesses? As the story progresses, to what degree do her traits become more or less prominent, and how does that work out for the character?

Andrea is a real estate agent, and she is a cunning real estate agent.  This single trait is both a strength and a weakness for Andrea.  It's a strength because she knows how to cunningly display a property for prospective buyers.  She will light a fire in the fireplace to make the house seem more welcoming, she will put special fragrances in the air, and she will even bring her dog if a possible buyer is a dog lover.  Andrea refers to these tactics as her "tricks."  She's a shrewd real estate agent, and it has made her successful.  I see that as a strength of hers.  
Unfortunately, Andrea's cunning and shrewd practices have started to consume her.  She becomes obsessed with the bowl's ability to help sell a house.  She reveres the bowl with superstitious fanaticism.  Even beyond that, Andrea begins to think of the bowl as a person that she has a relationship with.  

Could it be that she had some deeper connection with the bowl--a relationship of some kind?

As the story progresses, readers come to realize that Andrea is a broken, hollow, and empty woman.  Her marriage is intact in only legal terms.  She feels a closer kinship with the bowl than she does with her own husband.  As the story closes, readers finally learn why this bowl means so much to her.  

She had first seen the bowl several years earlier, at a crafts fair she had visited half in secret, with her lover.

The bowl reminds Andrea of her former lover.  He left Andrea when she refused to leave her husband for him.  The bowl is more than just a realtor's trick to her.  It's Andrea's way of trying to stay emotionally close to a part of her life that has long passed her by.  Her marriage is dead.  Her affair is broken, yet in her eyes, the bowl is still perfect. 

In its way, it was perfect: the world cut in half, deep and smoothly empty.

Andrea's devotion to bowl is perhaps her way of trying to hang on to something from a life that she no longer is able to have.  Andrea is very much like the bowl.  She appears perfect; however, she and the bowl are both ultimately hollow and empty.