Object, not subject.
The reason I didn't mentioned about #2 is because it's wrong. It's actually understandable, but considered as ungrammatical. See the difference between #2 and 3/5. 買い物をしたいです is the main predicative in #2, but it's not so 3 and 5. Actually, ~たり can be treated like a noun clause as same as ~たりするの/こと (e.g. 飲んだり食べたりはだめです, 読んだり見たりが嫌いです). This usage is from classical Japanese, so I recommend learners use the standard expression.