I only have a Saxo (UK) account for less than six months, so treat these answers as completely non-expert:
1. You can buy shares on any exchange that Saxo give you access to from a single account.
2. I presume you're not going to leave a lot of cash lying idle in a trading account, as opposed to actively invested. So your money will de facto be converted to whatever currency the shares are denominated in. If you are going to be buying shares denominated in dollars and your income is in dollars, it doesn't make sense to convert it to euro on the way into your account and back to dollars on purchasing shares. Saxo UK charge 0.5% for each currency conversion (and the same on selling the shares again). If you are able to transfer dollars, a dollar account sounds the most sensible since that is one conversion less, but check with Saxo if there are any gotchas.
This thread from the other day probably doesn't answer many questions in your case, but let me know if you find out anything that could be usefully added to it.