Can Lightlinky work only with the smartphone app?

No. LightLinky needs a point in your computer to send linkys. In our case we use an extension of your web browser.
What is a web browser extension?

It is a little application that is installed in your web browser, adding new features to it. For LightLinky, adds the ability to configure and send new linkys from your browser to your smartphone app.
Why do I need to download an extension?

LightLinky needs a place where you can write and send linkys from your computer. We have opted for this solution due to simplicity of process for users.
Can I use a normal desktop application?

Not yet, but we have plans to make it.
How can I send images to my smartphone? I don't see the option.

Select the image in your web browser and copy url image, like you will do with links. When you will send to your smartphone, it will detect the image linky, and will ask you if you want to download it.
Can I send images hosted in my computer (locally, not internet hosted)?

Not yet, but it is a feature that we are developing.