Check the faq here for information about buying a digital piano.

Yes, a digital piano can sound better than the sound on this recording that my phone produces. There are many digital pianos that are great instruments.

Weighted keys and 'hammer action' are important things to get touch sensitivity. Read some more about it. Now it sounds like you ask for a comfortable car, but you don't want to pay for suspension.

There is no way to determine how much time you need to play a piece. It depends on time practiced and the quality of your practice. It also depends on your goals, for that piece and for your piano skills as a whole. A teacher helps in this process. A method book can help to start.

A teacher might help you to have fun with the instrument too. However, there are many online resources to get practicing. Again, see the faq, it's great.

Note that It's the 'doing' that the pleasure gives and the small progress you make over time. It's not the songs you play. You have to find that out for yourself.

It's a great hobby, go for it.

Any instrument with a keyboard is a "keyboard."

A digital piano is focused primarily on replicating the sound and feel of a real piano. Most feature 88 keys with weighted action and high quality acoustic piano samples. There will be other sounds that may or may not be any good but the primary focus is piano. This is what you should get if you want to play piano music at home.

If you want to be a gigging pianist you may want to look into a stage piano. This is a digital piano that's designed to be portable and solidly built for gigging. A stage piano will usually not have built in speakers because they're designed to be connected to an external amp or PA. In addition to acoustic piano, most have good electric piano sounds and some even have good organs, synth sounds, and others although they are still primarily piano-focused.

If you want a wide variety of of sounds as well as samplers, arpeggiators, etc you might look into a workstation keyboard. A workstation with 88 keys and weighted action gives you all the features of a stage piano plus many more non-piano sounds and features. There are also workstations with fewer keys and non-weighted action if you prioritize low weight and portability over real piano feel.

If you want to design your own sounds from the ground up, a synthesizer will let you do that. There are many, many different types so I suggest doing a lot of research before investing in one if you decide to go that route.

Any modern keyboard will have MIDI capability which gives you tons of flexibility. There are even MIDI controllers which are keyboards which produce no sound on their own but can be used to control external hardware and software.

The model you mentioned is great for piano music. None of the other sounds are particularly great but they might suit your needs if you mainly just need piano. You could gig with it but it's really not designed to withstand that much wear-and-tear. It's designed to be an affordable, quiet, convenient alternative to a home acoustic piano, and for that it's excellent!