I wouldn't worry about that little bit of insect damage or the dieing leaves unless it starts overtaking the plants. I just cut the leaves off when they look bad to allow sun to the newer leaves. Old leaves dieing is perfectly natural.

Based on what I can see of the soil, if you're watering every couple of days, there is a good chance the issue is related to over-watering. The heater shouldn't be an issue unless it blows warm air directly on the plant. Also, if the plant is being allowed to sit in the water that exits the pot after you water (there is a drain hole, yes?) the odds that it is related to over-watering or a high level of salts in the soil go WAY up. Ideally, you would water the plant thoroughly over the sink and wait for it to stop draining. Then, grasp the pot and move it downward, then sharply upward. You'll note that forces water from the drain hole. You WANT that because the water you're expelling limits root function and is prolly causing the issues in your pictures. Then, return the plant to the empty cache pot and don't water again until the plant is nearly but not completely dry. Use a sharpened wooden dowel or skewer stuck deep in the pot as a 'tell'. When it comes out dry, not moist, it's time to repeat the watering process. You should also be fertilizing regularly. More on that if you're interested. Al

Without a doubt! Thrips. I have them and its a constant fight. They destroy all my buds. They are sucking insects and if you grow Gardenia's or Hibiscus, they are Thrip magnets. I'm considering cutting them down but my Gardenia is huge and beautiful. A good control is Spinosid (not sure of the spelling) . You have to spray every two weeks while the buds are forming. It is organic and safe to use. Thrips will attack all orchid buds. They love Vandas and Phals but will go after anything they can get their miserable mouths on! Sorry about your plant. I understand the frustration. I'm going through it too. Jane

Regular date palms make terrible house plants, so they will probably continue to decline, especially if you are not providing additional grow light for at least 12 hours a day. Pygmy date palms, on the other hand make nice houseplants because they tolerate lower light levels. They don't like the cold, though. So it all depends on what you have. Where did you get the seeds? Fertilizing any container plant is a must because there are no nutrients in potting mixes -- there are palm formulas you get get online.