I have a class that serializes a set of objects (using XML serialization) that I want to unit test.
My problem is it feels like I will be testing the .NET implementation
I agree with you that you will be testing the .NET implementation more than you'll be testing your own code. But if that's what you want to do (perhaps you don't trust the .NET implementation :) ), I might approach your three questions as follows.
Yes, it's certainly possible to test the writer without the reader. Use the writer to serialize the example (20-year old Bob) you provided to a MemoryStream. Open the MemoryStream with an XmlDocument. Assert the root node is named "MyObject". Assert it has one attribute named "Height" with value "300". Assert there is a "Name" element containing a text node with value "Bob". Assert there is an "Age" element containing a text node with value "20".
Just do the reverse process of #1. Create an XmlDocument from the 20-year old Bob XML string. Deserialize the stream with the reader. Assert the Name property equals "Bob". Assert the Age property equals 20. You can do things like add test case with insignificant whitespace or single quotes instead of double-quotes to be more thorough.
See #1. You can extend it by adding what you consider to be tricky "edge" cases you think could break it. Names with various Unicode characters. Extra long names. Empty names. Negative ages. Etc.