In [1]: 
import numpy as np
import numpy.linalg as la
In [2]: 
eps = 1e-2  # set to 1e-5, 1e-10

A = np.array([
        [1, 1],
        [eps, 0],
        [0, eps],
        ])

np.set_printoptions(precision=20)
print A
print np.dot(A.T, A)

[[ 1.                      1.                    ]
 [ 0.01000000000000000021  0.                    ]
 [ 0.                      0.01000000000000000021]]
[[ 1.00009999999999998899  1.                    ]
 [ 1.                      1.00009999999999998899]]

Problem 1 is ...?

In [3]: 
n = 5

A = np.random.randn(5, 5) * 10**-np.linspace(0, -5, n)
la.cond(A)
Out[3]:
60358.505352514032
In [4]: 
la.cond(np.dot(A.T, A))
Out[4]:
3643149168.4817424

Problem 2 is ...?

General bound: \(\operatorname{cond}(AB)\le \dots\)?