Shutter button cable for time-elapsed pictures?

Yes, it's called a cable release. Some DSLR manufacturers even call it a 'remote shutter cable'. You may even be able to connect your camera to your PC and use a remote software to control the camera....

Another option may be to set it to manual settings with a long shutter speed and use the delay timer setting (like you would for self portraits). When the timer counts down and fires the shutter, all vibrations probably stopped.

Another problem may be the stability of your tripod. Most consumer tripods are not really strong enough to hold a camera still in a light wind... or dampen vibrations. If your camera moves for a second or two after you touch it, your tripod is insufficient.

What you are looking for is a cable release, and unfortunately I am unable to find where any of sony's cybershot series uses this accessory. This is most common in use with SLRs. Some screw into the camera right near the lens mount, and some of the newer cameras have cable releases that are either I-red, or have a prong such as that which is on the end of a set of earphones that releases the shutter through an electronic signal. Your best bet is going to be to call Sony with your camera model number and they will be able to tell you what accessories are available for that model.

A cable release is manual, and doesn't really apply to modern, digital cameras anymore. As previously suggested, a remote would be the best alternative.

If that's not available, there is a trick you can use - the self timer. While not as precise, because you're not snapping the image at the instant you set it, it does do away with camera shake from depressing the shutter release button.

The power button is another issue though, since you'd still have to turn it on each time it powers down.

A poor workman always blames his poor work on his tools. There is nothing wrong with your camera. The shutter won't work because you are trying to focus too close and Nikon built that fail-safe into the camera so you won't take a bad picture and blame the camera. For decades camera manufacturers have built their cameras to be as fool-proof as possible. Their goal is, was - and always will be - to insure that the casual user gets as many good pictures as possible to make the user happy and build brand loyalty. The 18-55mm lens is designed to close focus to 11 inches. Get any closer and you will have nothing but a blurry mess. You have to know the limitations of your equipment. This is best learned by taking time to READ & STUDY the Owner's Manual for your camera and lens. If macro photography is one of your interests then invest in a macro lens like the AF-S VR Micro Nikkor 105mm f2.8G IF-ED.

You should look and see if there is an electronic remote for that camera, it works just like a tv remote, some cameras have that capability some don't - worth checking. With one of those there's no need to touch the camera at all and nothing to attach.

Thats called a cable release.