Explanation
The modified duration of a perpetual bond is given by 1/i, where i is the yield. Since I is a positive number greater than zero, it means the modified duration of a perpetual bond is not infinity. Also note that this formula makes no reference to the coupon rate at all - in other words, the modified duration of a perpetual bond is independent of the coupon. The Macaulay duration of a perpetual bond is given by (1+i)/i, again a formula devoid of any references to the coupon. The price of a perpetual bond can be calculated as Coupon/i, which means that a 15% coupon perpetual bond will be priced at exactly 3x the price of a 5% coupon bond.
Based on the above, statement I is incorrect, but statement II is correct. Statement III is incorrect because Bond A will be priced at exactly 1/3rd the price of Bond B, and not less than that. Similarly, statement IV is incorrect as well.
Intuitively, a perpetual bond is nothing but a perpetual annuity, which will have a present value equal to 1/i * cash flow (which in this case is the coupon). The coupon rate only sets the cash flow dollar amount - there is really no difference between the two bonds described in the question except that buying one of Bond B is economically identical to buying three of Bond A.