In special relativity is there a simple case in which we have length contraction but no time dilation and vice versa?
The quick answer is "no", in the sense that time dilation and space contraction are simply ways of discussing aspects of the Lorentz transformation, which implies them both. However, in order to answer more fully, I think we should state your question more fully, and perhaps it could be stated "is there a phenomenon that can be attributed to time dilation in one frame, which could not be attributed to space contraction in some other frame?" For example, atmospheric muons arrive at sea level owing to their long life (in Earth rest frame) or owing to the small distance through the atmosphere (in muon rest frame).
I think we can say that the answer to the more fully stated question is no, by the following argument.
Time dilation is the observation that the time interval between two given timelike separated events depends on the inertial frame in which it is measured, such that the frame where the interval is longest is the one where the events have no spatial separation. Let the events be A and B, and suppose there is in the vicinity of these events some large rigid inertially moving object R, wide enough to extend between A and B. Suppose also that there is a clock which travels at constant velocity from A to B, and makes a mark on R as it arrives at each event. We have now caused our generic time dilation example to be just like the example of atmospheric muons. The fact that the clock completes the journey from A to B while its hands rotate by some given angle can be ascribed either to the fact that the distance between the marks on the rigid object is small in one frame, or to the fact that the clock runs slow in another frame.
Finally, if you don't want to have a rigid object R, note that the very words "inertial frame" and "distance" are shorthand ways to refer to what such an object will be like; and similarly if you wish to remove the clock then note that the very words "observe" and "time interval" are shorthand ways to refer to how such a clock will behave (no matter how it is constructed).
No. If time dilation is the effect on B's frame obsered from A's frame then length contraction is the effect observed on A's frame from B's frame. You cannot have one without the other as they are different perspectives of the same phenomena.
Note: To be clear it is entirely possible for a single observer to experience/measure both time dilation and length contraction at the same time.
It depends what you mean by "length contraction". But here's one scenario that, depending on your meaning, might apply:
Take a stick that is (in your frame $E$) 1 meter long and initially at rest. Now contrive (say by applying appropriate magnetic fields) to accelerate both ends of the stick in the direction the stick is pointing, according to some time-varying acceleration $a(t)$ (all as measured in frame $E$). Assume you manage to do this in a way that doesn't break the stick.
Clearly, as measured in frame $E$, the length of the stick never changes. But a clock mounted on the stick will tick slower (again, as measured from frame $E$) once the motion gets underway.
You observe length contraction in the direction of the velocity. If an object moves away from you, you don't see any length contraction, but you do notice time dilation. Conversely, if you don't observe time dilation, the Lorentz factor is 1 and your relative velocity to the object is zero. Then there is no length contraction. John