How to prevent bed movement's noises when having sex?

Sounds from beds moving, whatever cause, seems to stem from three different internal sources and one external source:

• The legs – This can be solved by having pads under the legs of either rubber, some soft substance. Plenty options exists at any furniture shop
• Wood joints – If there are wood touching wood, when moved these will screech. Some common options to avoid this are to glue pieces together (not always viable for a bed), add something between the joist (like a fabric or thin rubber sheets) or tightening the screws. The latter will usually only postpone the isssue
• Mattress vs frame – In the older beds with planks across the bottom, or in newer beds (like the sommier beds) with the entire mattress resting upon a frame (or on hinges), there could be sound generated between the mattress and the frame. To avoid this sound the solution is still to add something inbetween the two elements making the sound. Maybe you could add a rubber adhesive/band on top of the bottom frame, if this is making the sound?
• Bed touching walls – As mentioned elsewhere, if the beds bounces into walls, dressers, or other furniture this sound might propagate to the building structure. In this case, simply move the bed further away.

In short, you need to locate the source of the sound and address it specifically. And yes, the sound from bed activities in apartments can be heard through floors (or building structure). I've been hearing in it in some of the apartments I've lived in, and it does create an awkward tension when meeting the tenants later on.

The amount of sound carrying through varies greatly though, depending on what kind of building and age and other factors, so trying to devise some test to discover what carries through could be worth indulging into. Not sure if one of you can visit the neighbor underneath, and the other can "clean the aparment" and jump in the bed, move the bed, and so on, whilst you are listening to hear any of it.

Rubber furniture cups like these rubber furniture cups. We have the square ones for our bed which solved the problem for us! :) I like them a lot and they really work well on hard flooring.

SHEPHERD RUBBER furniture cups

Instead of ordinary bed, you can buy and use a beanbag bed.

Such a bed won't make any noises or squeaks as far as I can tell, though just from common sense, not personal experience.

Example for such a bed:

There are also beanbag sofas that turn into a bed.

The product for this is stick-on pads of felt or rubber under each leg. For an improvised solution, lay a carpet or mats/wads of any somewhat soft material under each leg (a tightly folded burlap sack under each leg would work). Other approaches that can be combined with padding the legs are: Increasing the mass of the bed itself (this increases friction and inertia), and reducing the horizontal components of your combined centre-of-mass velocity (this reduces sideways forces on the bed). Also, do not touch the wall, any force on the wall results in an equal and opposite force on the bed, resulting in dragging.

If the bed also rocks (the ends of the legs are not coplanar), the legs need to be adjusted or propped with objects of differing thickness.

Well most of the sound is produced from wood-to-wood joint. Unscrew you bed apart, then use ordinary soap (use it dry) to rub it onto touching surfaces. Screw it back together tight, then screw her :) Should work for some time.

If its a wooden bed, undo each joint in turn and rub beeswax on the inner surfaces of the wood that meet, then tighten up really tight again. Worked for us. Also consider putting some small pieces of carpet, or mat under each leg as stated above.

You could use some stick-on for the bed and door-stopper for the door problem.

Tennis balls colored to match the legs or frame should damped any noise.
If they slip and slide, maybe put them on top of nonslip fabric.

Grab your socks with holes in them, and put them over the legs. This might help enough to do the job.