LINQ to Objects is your friend. For the first:
bool allSameX = list.All(t => t.x == list[0].x);
Test firstTry = list.First(t => t.z == "try");
Test firstTryOrNull = list.FirstOrDefault(t => t.z == "try");
The first one depends on there being at least one value of course. Alternatives might be:
bool allSameX = !list.Select(t => t.x)
.Distinct()
.Skip(1)
.Any();
In other words, once you've gone past the first distinct value of x, there shouldn't be any more. One nice aspect of this is that as soon as it spots the second distinct value, it will stop looking - as does the first line (the All version) of course.
LINQ is wonderfully flexible, and well worth looking into closely.
EDIT: If you need to do the latter test ("find an element with a particular value for z") for multiple different values, you might want a dictionary or a lookup, e.g.
// If there are duplicate z values
var lookup = list.ToLookup(t => t.z);
// If z values are distinct
var dictionary = list.ToDictionary(t => t.z);
Without some pre-work, there's no way of performing the queries you want without iterating over at least some of the list.