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We present new upper and lower bounds on the minimum distance of certain generalized bicycle (GB) codes that go beyond the capabilities of techniques based on classical codes. These bounds are shown to capture the true minimum distance for specific subclasses within the GB family. In particular, we demonstrate the existence of two highly degenerate GB code families with parameters $[\![d^2+1,2,d]\!]_2$ for odd $d \geq 3$ and $[\![d^2,2,d]\!]_2$ for even $d \geq 4$, both having qubit connectivity of four, similar to that of surface codes. For the odd-distance family, we analyze the structure of low-weight logical Pauli operators and establish the existence of a fault-tolerant logical CNOT gate between the two logical qubits, achievable via a simple relabeling of data qubits. Next, using nearly self-inverse polynomials, we extend our results to construct an infinite class of connectivity-six GB codes with an improved rate. For GB codes based on these polynomials, we further present a technique to realize transversal logical CNOT gates via data-qubit permutations. We further construct syndrome extraction schemes for our connectivity four GB families that avoid reductions in minimum distance due to circuit-level faults propagating from check qubits to data qubits. Finally, we numerically evaluate the logical error rate performance of the mentioned connectivity four families of GB codes under circuit-level noise, obtaining thresholds comparable to those of rotated surface codes.