I'm new to TFS API. I have looked around but found incomplete solutions to my problems.

What I want to do is search for a specific test case by its custom field "Extended ID" = "EXT:123" in Test Plan with Id = 104, and Test Suite with Id = 1455. Maybe there are 2 parts to this question.

1) Is TestSuiteID unique throughout the entire test plans within a project? If not, then I would have to search for a specific test suite by digging down the hierarchy "Project"-->"TestPlan"-->"TestSuite". How do I do that?

2) Could anyone kindly show me how I could search for a specific test case by its custom field "Extended ID" = "EXT:123" within a specific test suite?

So far, I have

Code:
string tfsUrl = "http://tfs:8080/tfs/DefaultCollection";
        string projectName = "MyProduct";
        int testPlanId = 111;
        int testsuiteId = 2222;
        int extendedId = "EXT:123";

        ITestManagementTeamProject proj = GetProject(tfsUrl, projectName);

        var tps = proj.TestPlans;
        var tp = tps.Find(testPlanId);
        // How do I find a Test Suite within a Test Plan? 

        // Searching for a Test Suite ID within a project works only if its unique throughout all test plans within the project
        var tss = proj.TestSuites;
        var ts = tss.Find(testsuiteId); // This works 
        var tcs = ts.AllTestCases;

        foreach (var tc in tcs)
        {
            // I want to find a test case with custom field "Extended ID" that matches "EXT:123"
            if (tc.??????? == extendedId) 
            {                    
                var testResults = proj.TestResults.ByTestId(tc.Id);
                // How do I update its test result?
Thanks in advance!!