To expand on my comment:
Coconut milk, chips/shredded, flour, water, and oil are completely different things though they come from the same nut.
Coconut Oil to me seems the most interesting as it both contains safe saturated oils, and MCTs, which are very beneficial and easily convert to ketones. They seem to be very useful for folks whose brains can no longer handle glucose properly (i.e. Alzheimer's/aka "Type 3" diabetes as per Dr. Bruce Fife et al.)
MCTs are very useful when you're trying to adapt to a low carb diet if that's your goal, and can raise your heart rate, burning more calories as well. You can easily overdo it by doing nothing but coconut oil and coffee, which will completely kill your appetite, and then you get a cortisol spike which is necessary to kick in gluconeogenesis to provide glucose for blood sugar and brain cells.
MCTs are can also attack some viruses and some bad bacteria without harming the good bacteria. Depending on your gut flora, this can cause an imbalance, so of course you can get gut issues.
Another benefit is that since the MCT oils don't get processed by the same pathway as other oils, you can give your liver a rest, so it can heal, and build up bile levels.
When you're first starting out on coconut oil, it's dose dependent. If you take too much at once (for some people it's as low as a tablespoon), you can easily get the runs because your digestive system cannot produce enough bile to deal with the amount of fats coming in. You'd get the same issue with butter or any oil if you over did it. Over time, you can do several tablespoons a day, and not have this issue.
So if you overdo the oil, yes, you'll get the runs. Once you get used to a larger dose of oil, you won't. Some canned coconut oils have things like guar gum in them, preservatives, etc. Your constipation issues sound more like either a lack of magnesium, or lack of stomach acid/bile. I've never heard of coconut causing a Herxheimer reaction.
Then, some people have actual, real allergies to coconuts, in which case they obviously should be treated the same as wheat. They might not have the same issues to just MCT oil, so YMMV.
Coconut meat which you can find in wraps, frozen shredded meat, milk, chips, shreds, flour, (the last being defatted meat) do contain loads of fiber, and those are sugars which we cannot digest, but depending on our gut flora, gut bacteria can. As you know, gut bacteria can also digest the sugars in beans, which cause gas, well, in the same way, depending on what gut flora you have, the same might happen. Or you might have different bacteria, the beneficial kinds which can convert the fibers into fats, and all will be well.
Coconut milk in a can, I love the stuff, and I'm probably both a coffee and coconut milk addict, but you can have a reaction to the preservatives, or if Guar gum or other additives are added in, to that, or have issues with the BPAs in the can lining.
I've recently seen "Silk" brand coconut milk in stores, but it has some nasty stuff in such as carragean which is a suspected carcinogen, but they add it in so they can sell a water down product that still has the same thick mouthfeel as the stuff that's full fat and not watered down. (I've also found that things like "organic chocolate milk" aimed at kids have that stuff in it, which, if it turns out that carragean is indeed a carcinogen should be outright criminal.)
So it depends widely from brand to brand whether it will agree with you.
Coconut water is a wonderful source of potassium and a few other minerals, so it tends to be a great electrolyte/recovery drink, so you can skip the artificial crap in things like gatorade. Also, it seems that in potassium pills are for some reason limited to a very small amount (99mg), so you'd need 6-7 pills to match what's in a single box of coconut water. You could of course get it from cucumbers or bananas, but unless I've just had a very intense workout, I'll stay away from bananas.
Keep in mind that your gut flora can and does change depending on what you eat, especially the kinds of carbs/fiber you ingest, and what fermented foods, probiotics/yeasts, you ingest. Things like coconut oil, oregano oil, antibiotics will absolutely affect them, so while you might have issues with coconut today, you might not tomorrow (or vice versa). Unless it's an actual allergy, in which case, it's a whole different ballgame.