That's the thing, these type of buttons don't necessarily need to be physically stuck down, but the contact inside may be sufficiently made either physically or bridged by foreign matter, as to create a false press. I have seen that a couple of times when kids have had juice etc on their hands and pressed the buttons. It doesn't happen immediately, takes months, long enough after the incident that nobody remembers it happening. However, in saying that, removing the batteries would certainly rule the hand control out of the equation.
It could still be a firmware issue, or a power supply issue, or just a bad regulator or dry solder joint somewhere on the main board. You mention that the picture is good throughout the whole blinking thing. Are you certain that the backlight is not flickering in any way or that the picture is totally unaffected when it starts with the blinking LED and sound popping in and out?
If you feel confident in your ability to repair the set, your first action should be to take loads of good quality, close up photos and post them here so that we can at least attempt to identify the chassis and board manufacturer. Otherwise, I would suggest, as this seems to be your main/ only TV, that you take it to a half decent TV Engineer and let him have at it. It may cost a bit more to get him to repair it, but better that than either a new TV immediately, or worse, you spending needless money buying parts that are not faulty and then having to buy a new TV anyway.
The sound coming and going certainly hints at a power supply issue, but as the audio circuitry is usually a lower current demand, other areas of the set often show signs of distress first. That doesn't rule out a power supply issue by any means, but it may, given other clues, point you in a different direction.
Do you have some basic test equipment and tools to hand? Do you know how to use a Multimeter? Multimeter? Soldering iron? Desoldering braid? Mixture of small, insulated, screwdrivers, long nose pliers, small side snips etc? If the answer is no to any of these questions, then you really are between a rock and a hard place with this and I would say a TV Engineer would be the route to take.