Yes, it can have many manifestations. My husband has had what he thought was psoriasis all his life. He has also suffered from bloated gut and many aches in it. He would periodically get diarrhea. After ultrasounds nothing was found and he can't get in for a colonoscopy until after April. He did some research on his own and thought he would try a gluten free diet to see. What a difference it made! Much of his pain is gone and some of his bloating (fatty foods and too much alcohol do contribute to his issues) and his psoriasis has almost cleared up without the use of any of his medications. His bowels are working better and we did a test last week where he ate bread and the diarrhea came back.
Something to be aware of though is that if you do change your diet before you have tests and biopsies that they will not come back accurate.
Gluten intolerance can present with a variety of different symptoms, so it would be hard to say what 'standard' is. Some people have pain like GERD, others have abdominal pain. Some have little to no pain, period. Some get rashes, some don't. Some people have 'brain fog,' others nada. Some get so tired they feel like if they don't lay down they'll fall down. Other people feel slightly fatigued. Etc., etc., etc.
Some peopel can be diagnosed via blood tests, but doc's prefer a biopsy via endoscopy. The problem is neither of those tests is 100 percent in my opinion. There's a very good fecal antibody test coupled with an allele test that will tell you whether or not you carry the propensity (basically genes) that suggest you may carry the trait. I believe a couple of labs do that testing now, but you can read about the whole situation at enterolab.com. My husband was diagnosed using the fecal antibody test when nothing else indicated he was gluten intolerant.