Ok, from what I can gather from your post, I'm assuming that all these issues are in sandbox, and you haven't yet made an attemp to deploy the code to production.  I mean, the existing tests are failing in sandobx, right?  If this is the case, then try removing YOUR trigger and test code, and then try running the existing tests.  If they still fail, that means you will have to fix the already written test code (that you did NOT write).  There could be numerous reasons why the tests that had earlier passed, fail now.  Here's a couple I can think of from the top of my head:

• The test code does NOT create all the test data, and instead uses hard-coded IDs from Production.  The test data being used in the test code should always be created in the test code.  Actual data should never be used for testing.  All data/ records created in test code are mock records, and are not persistent.
• The test code was not written to cater for large sets of data.  At the time the triggers were deployed, there was a small set of data, which the test code processed correctly.  But now, with bigger data set, the test code hits governing limits.

There could be others, but these are generally the two main reasons for such situations.  Hope this helps!