Do I press the sustain pedal when there is a tie going accros serveral notes?

In the picture there is a Tie going across 16 sixteen notes (lol). Do I press the Sustain pedal?

Actually it's not a tie, it's a slur or phase mark. If it's a slur it means that you should play the notes legato, basically it means that there shouldn't be any pause between the notes or it should at least be as short as possible. Dunno if I'm describing it right but you can check Wikipedia otherwise: http://en.wikipedia.org/wiki/Legato

As for the pedal marking you press down the pedal where the Ped is marked and you release it when there's a "star" (dunno what it's called).

Hello!

If you are asking yourself this question, you should probably not be playing this piece.
No, you should not press the pedal during the whole slur, obviously, because it sounds bad. Things that sound bad are... bad.
Also, the pedal mark written on your score is not Chopin's original (he didn't write pedal marks for this piece), which doesn't mean that you should not use pedal; it just means that the use of pedal is your own responsibility.
It's ok to use scores from the internet as long as you play pop songs or easy pieces. But for Chopin's etude you should definitely look for Urtexts such as Paderewsky edition or Henle. But seriously, if you're asking this question about pedalling, there is about 99% chance that this piece is way too difficult for you.