A. While childcare and education make up a considerable part of the total cost for child-rearing in a child's first few years, overall, it is the costs of additional housing, which largely stems from the need to have an extra bedroom, that makes up the lion's share of the expenses incurred throughout the child-rearing period.
“it is the costs… that makes up” – subject-verb disagreement
“cost of child-rearing” is idiomatic
“which largely stems” refers to “additional housing” rather than “the costs” as it should
B. While childcare and education make up a considerable part of the total cost of raising a child during his or her first few years, additional housing, stemming largely from the need to have an extra bedroom, makes up the lion's share of the expenses incurred throughout the child-rearing period.
Seems free of errors since modifiers are placed correctly as far as I know.
C. Even though childcare and education make up a considerable part of the total cost of raising a child during his or her first few years, the costs of additional housing, which in most cases stems from the need to have an extra bedroom, are the ones that make up the lion's share of the expenses incurred during the child-rearing years.
“are the ones” is a superfluous phrase. without it the sentence would read much more academic “the costs of additional housing… that make up”
D. Childcare and education make up a considerable part of the total cost for the first few years of raising a child, but overall, the lion's share of the expenses incurred throughout the child-rearing period is that spent on additional housing, needing in most cases to have an extra bedroom.
Adverbial modifier at the end of the sentence doesn’t make sense with its alleged subject “lion’s share of expenses”.
E. The costs of childcare and education make up a considerable part of raising a child during his or her first few years, yet the lion's share of the expenses incurred throughout the child-rearing period are spent on additional housing, mostly because of the need to have an extra bedroom.
“expenses… are spent” is illogical. “expenses” can’t be spent; they can be made. “the lion's share of the expenses” - have never seen “share” be tested by GMAT. Probably “share” is a singular noun as used in choice D.
So B