Directional stability is affected by the side areas such as the vertical stabilizers and lateral stability is affected by dihedrals. Is there any relationship between lateral and directional stability as a roll result in a pitch and most aircraft are configured to have more lateral than directional.
For stability in roll, yes there is a direct relationship. To achieve good stability in roll from dihedral effect, directional stability must not be too strong.
This is because dihedral effect requires sideslip to work. But directional stability (weathervaning in yaw) tries to minimize side slip. If the plane is bumped into slight bank while flying along, some sideslip must be allowed to develop before the weathervaning forces in yaw take effect to get the tail aligned with the nose in the airflow.
So to have good lateral stability, you need directional stability characteristics to be degraded just enough to ensure a slight lag in the directional weathervaning tendency, because it's the lag that allows a bit of sideslip to develop, which is what gets dihedral effect to work.
Tail volume calculations (area x arm) take this into account in vertical tail sizing. The vertical tail volume should be large enough to have a strong weathervaning effect, but not strong enough to prevent a small amount of initial sideslip to develop when banked by a disturbance.
If you do make the vertical tail too large, you will have fantastic directional stability as the plane instantly weathervanes into the airflow when disturbed in the yaw axis. But without a slight lag in the weathervaning tendency following a yaw disturbance, when a lateral disturbance occurs, no side slip develops, and dihedral effect can't work.
The result is the plane becomes laterally neutrally stable, or maybe a bit unstable, as any roll disturbance that banks the plane causes it to weathervane to stay aligned in the airstream instantly, before sufficient sideslip can develop to get dihedral effect going. If you leave the airplane alone while it's doing this, you'll find that every lateral bump that causes the airplane to bank a little bit causes it to spiral toward the low wing. In other words, too much vertical tail volume causes a spiral instability or spiral diving tendency.
And obviously, too little tail volume just results in a plane that slithers around the sky and is unpleasant to ride in, and if you don't have a yaw damper, your feet have to do the job to keep the thing straight directionally.
I flew a small float plane one year doing float ratings, a Fleet 90 Canuck, that was at that other extreme of insufficient directional stability. Its certified float installation did not include an additional ventral or dorsal fin to account for the destabilizing effect of the floats (added side area forward).
Its directional stability on floats was only mildly positive, and the nose would drift all over the place in bumpy air and your feet were going constantly. But its lateral stability was great because lots of sideslip could develop when it was bumped into a bank, so its dihedral effect and lateral stability was very strong.
a roll resulting in a pitch
Yes, there is a relationship here, as a roll results in loss of vertical lift, and the plane will sink. The sink should pitch the nose down.
But dihedral itself does not fix roll instability in a cross wind gust, in fact it makes it worse. Dihedral must be balanced with a "keel" below the center of gravity. This balances side force around the Center of Gravity. If the plane is deliberately rolled, dihedral also helps it upright if the rolling input is removed and the roll is only slight.
The "book" says sideslip must develop, but sinking a little bit is enough to "activate" the righting dihedral effect. The combination of keel (to prevent rolling in a sidewind gust) and the dihedral (to help right any unintentional roll) makes for the classic "high wing" trainer design.
Excessive directional stability in the form of a large tail does introduce "spiral instability", but remember, the torque of a tall, upright tail, especially with coordinating rudder applied, also favors a "righting", or dihedral effect. Changing the twin tail of the B-24 Liberator to a single upright vertical fin on the PB4Y-2 Privateer greatly improved its flying characteristics.