The abs_ function is part of the Gurobi Python API and does not work in an LP file. (When your LP file is read, the solver interprets your C3 constraint as a linear constraint with four variables with names abs_(v0, v1), abs_(v1, and v2).)

Also, note that absolute value constraints can only take the form var1 = abs_(var2). So you will have to define some auxiliary variables to model the constraint above.

In the Python API, you could model the constraint C3 as follows:

a = model.addVars(4, name="a")   # auxiliary variables
model.addConstr(a[0] == v0 - v1)
model.addConstr(a[1] == v1 - v2)
model.addConstr(a[2] == abs_(a[0]))
model.addConstr(a[3] == abs_(a[1]))

model.addConstr(a[2] + a[3] >= 10)

In the LP file, you can see the absolute value constraint in the General Constraints section:

General Constraints
 GC0: a[2] = ABS ( a[0] )
 GC1: a[3] = ABS ( a[1] )

y=|x1-x2| for binary variable is the same as y = x1 xor x2. This can be written as:

y <= x1+x2 
y >= x1-x2 
y >= x2-x1 
y <= 2-x1-x2

(see here).

When x1,x2 are continuous variables between 0 and U you can write:

y1-y2 = x1-x2
y = y1+y2
y1 <= delta*U
y2 <= (1-delta)*U
y1>=0
y2>=0
delta in {0,1}

In many cases this can be simplified.