Every car has a flywheel, even an auto transmission one, and when you release your foot off the gas, the flywheel will start to slow down. It will slow down quickly until under 1000rpm when the ECU finally decides it is too low and tells the electronic throttle to give it a bit more gas to keep the RPM up. If you get someone to observe the RPM dropping, it should dip below 1000rpm but still stay above 700rpm or so. Old auto transmission fluid can also contribute to this. Because of the way the torque converter works by using the transmission fluid to create friction in the torque convertor as the RPM increases, old fluid can have more slip which contributes to it.
If the transmission fluid has not been replaced in the past 2 years or you have no records of it being done, I recommend that you do it. Use only Honda auto transmission fluid (ATF).
If it sounds like a diesel, and it is worse when it is cold, then you need to get the valve clearance of the engine checked and adjusted if required. This is part of the service schedule and should be carried out every 2 years or 40,000km.
E10 itself is ok, but the majority of E10 out there is 91 octane, which is way below the minimum requirement of 95 octane. As long as the previous owner did not thrash the car on 91 octane, the engine will be fine. The ECU listens for knock from using too low an octane and adjusts the air/fuel mixture accordingly to protect the engine.
It is actually a feature so that the car slows down using engine braking when you are going downhill and you will end up using less of your brakes (and making it last longer). The ECU learns your driving style and changes to suit, so if you find it annoying, just use your brakes more when you are going downhill and after a while it should remember that and use less engine braking.
When this happens again, stick it into semi-auto mode and shift it into the next gear. Do this for a few days and see how it goes. Also, it shouldn't vibrate that badly. Change the auto transmission fluid and see how it goes.
It's because you don't have a revving engine and a revving gearbox. So at least you know where the issue is coming from. In addition, check the accessory belt. There is an auto tensioner for the belt behind the power steering pump and it has a triangle which points at a large rectangle and a smaller square on it. The triangle should point at the large rectangle or be in between the rectangle and the square. If it is pointing at the small square or even beyond that, then the belt needs to be replaced.
Yes, it is replaceable and is actually a wear and tear item that should be replaced after a few years. I buy mine from the parts shop for about $60 and replace it myself. The belt width and length is 7PK1760 (7 ribs, 1760mm circumference). If you can't find this one, then 7PK1750 is fine as well.