T-class range and the Far Eastern ambush mission

Was the T class designed for Mediterranean service, and did it have a tendency to dive suddenly without warning?

No: it was a long-range Far Eastern successor to the O, P, and R boats, while the loss of HMS Thetis was an accident rather than a class-wide diving tendency.

Two premises in the question need correction: the T class was intended to succeed the O, P, and R boats for Far Eastern service, and it did not habitually dive without warning. HMS Thetis suffered a major accident during trials, but this was not treated as a class-wide tendency. The boats were built around an ambush role against Japanese traffic. Their roughly 8,000-nautical-mile surface range and 131-ton fuel load enabled distant patrols from limited bases. The U class was less than half the size and optimized for nearby Mediterranean or North Sea waters. The T class instead carried a heavy forward torpedo battery for waiting near a channel, firing a concentrated salvo, and escaping.