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Piano concerto

in Piano

I am not a great composer of any kind, but I thought that I could try my hands on a piano concerto and currently I have a few questions.

1. How do chords really work? When the piano has the melody, how should I construct an accompaniment following it?

2. How am I supposed to write condensed arpeggios? If I just use chords it doesn't sound right. If I use intervals I don't know which one to use. Is it a combination of both or is it simply one? (If it's a bit confusing look at Rachmaninoff's piano concerto no 2's 1st page)

3. How should I go on attempting harmonies? I have a theme and melody on the piano and strings most of the time, but how should I add pieces to it? I could just use chords in accordance with the melody which I have done. I could also another line of melody making a bit of counterpoint, but anything else?

I've been studying Tchaikovsky's 1st for my own piano concerto. There's a lot of things he wrote that I don't quite understand. So if you have any more suggestions than my questions, feel free to type it up.

Would appreciate some help finding classical piano piece

in Piano

I remember playing this piece as a child. It was definitely a waltz, although I don't really know who may be the composer or what the name of the piece.
I hummed this piece and made an approximate sheet for its beginning. Please, take a look here:
This was made using a virtual piano so I just made a rough representation of the melody. I may be not fully correct but the general tune should be the same. Please take into a consideration that I couldn't draw bars where it is appropriate to do so and that sharp note doesn't stay sharp until natural sign (becarre) is applied (this program is really simple so it doesn't have natural signs). For example if there are C# and then C immediately after it then this second C is a natural (flat) C and not a sharp C.
I would greatly appreciate if somebody recognizes this melody (I'm sure this is a waltz) and helps me to find its name and composer.