Fruit is pretty easy: berries are ok for the occasional small dessert, be they straw-, rasp-, black-, blue-, mul-, or others. Pour some heavy cream over them, and you've got a high-fat dessert with just a few carbs. (If you don't do dairy, just eat the berries by themselves.) They're high in fiber, so most people here wouldn't eat a lot of them, but if you're not eating grain and high-fiber vegetables, your total fiber intake will be very low anyway.
Pretty much all other fruit is out, especially dried fruit. All the other fruits you mentioned are very high in carbs. Some melon isn't too high, but it's high in fiber and fructose, so it's best avoided when you're trying to heal from past damage (which goes along with losing weight). (Plus, I find it's a lot easier to stop at a 1/4 cup of berries than a 1/4 cup of watermelon, since I grew up eating watermelon by the thick slice. It doesn't seem worth the trouble to cut into a melon to get a low-carb serving.)
On vegetables, most of the green ones tend to be low-carb, especially if they're leafy: broccoli, lettuce, Swiss chard, spinach, cabbage, mustard greens, Oriental greens, etc. Green beans are low, if you don't think they're too legume-y. Some exceptions to the 'green' rule would be peas and lima beans -- since they're the starchy seed part, they're high in carbs. (Snow peas with undeveloped peas in the pod are low though.) Cauliflower is borderline, as are tomatoes: a slice of tomato on your burger now and then may be fine, but tomato-heavy recipes or tomato-concentrates like sauces generally aren't. Some root vegetables are low also, like radishes and turnips. Carrots are high -- if people squeeze the juice out of something and consider it sweet enough to drink, you can figure it's high-carb. Summer squash and its watery relatives like zucchini and cucumbers are low; the solid and heavy winter squash like butternut and pumpkins are fairly high.
Note that I'm not addressing the paleo-appropriateness of any of these things, except the mention of fructose. Once you lose the weight and get healthy, you may be able to switch back to starchier vegetables and be choosier about avoiding things like fiber or nightshades if you wish. In the meantime, you might want to mix things up -- don't just eat the same one or two vegetables all the time, in case you have a problem with them -- or just keep your fruit and vegetable consumption very low.