Well, you don't strictly need an account in dollars to invest in US securities (but you are more exposed to exchange risks) while it is handy for other kinds of investments in US dollars.

As riffraff said, Fineco is one of the very few banks (maybe the only one) that gives the possibility to have foreign currency accounts (linked to a "main" Euro account) at a "reasonable" cost, which is a good thing to have if you deal a lot with the US (both with receiving US dollars and spending in US dollars), but that is not needed for "normal" activities.

The "conto in divisa estera" (that quite a few banks have) is basically aimed to business entities (like firms doing import/export) and is usually rather expensive, or - to be more accurate - has very high fixed costs.

The alternative in the case of someone receiving US dollars may be some form of accounts with credit/debit cards issuers, but - again generally speaking - they - besides offering less possibilities (and not being "properly" a bank) tend to have low fixed costs but higher commissions/fees on single operations.

I do. I have an account with Interactive Brokers and I can easily deposit EUR. I can buy US stocks and options in USD and IB takes care of all the details (internally, it shows up as a short USD position).

Investing from the banks themselves it's also possible but it involves higher fees. Using IB has the disadvantage that I have to do the taxes myself.

I've yet to figure that out. Personally, I'd love to be able to easily invest in a vanguard fund, or even just have the ability to put money every month on index funds, but so far from what I've seen doesn't seem to be possible.

You can easily buy everything you want from Interactive Brokers. You can deposit EUR and do the transactions in other currencies. The interface is old and complicated, but very powerful. You will have to do the taxes yourself though.