Shifting and adding the columns of two lists

a = {{-2, 1}, {-1, 2}, {0, 1}, {1, 0}, {2, 0}}
before = a[[All, 2]];
a[[All, 2]] = RotateLeft[a[[All, 2]], 3];
a[[All, 2]] = a[[All, 2]] + before;

Now a is your c

Step-By-Step

 (a = {{-2, 1}, {-1, 2}, {0, 1}, {1, 0}, {2, 0}}) // MatrixForm

before = a[[All, 2]];
a[[All, 2]] = RotateLeft[a[[All, 2]], 3];
(b = a) // MatrixForm

b[[All, 2]] = b[[All, 2]] + before;
(c = b) // MatrixForm

SubsetMap[RotateRight[#,2]&, a, {All,2}]

{{-2, 0}, {-1, 0}, {0, 1}, {1, 2}, {2, 1}}

%==b

True

For Part 2, see this recent question

'Borrowing from the neat answer given by David Keith:

Transpose[{a[[All,1]],(a+b)[[All,2]]}]==c

True

Alternatively:

a+ArrayFlatten[{{0, List/@b[[All,2]]}}]==c

True

SubsetMap[Reverse, a, {All, 2}] == b

True

SubsetMap[# + Reverse @ #&, a, {All, 2}] == c

True

Or get b and c in a single step:

Rest @ FoldList[SubsetMap[#2, #, {All, 2}] &, a, {Reverse, # + Reverse @ # &}] == {b, c}

True