Horizon terminal. At least it was last week. Your embarkation form will confirm nearer the time.
Hamburg - yes, although you don't need it and it was pay extra per person. You can walk off the shop and it's 10min walk to the ferries. Hamburg by ferry was great fun! Have a Google.
Zebrugge - yes, but only to centre of Zeebrugge, and again, pay extra. If you want Bruges, take the FREE shuttle yo the cruise terminal laid on by the port (not MSC) and there is a €20 coach (return) to take you to Bruges. Journey took about 20mins and was very easy.
Rotterdam - unfortunately this stop was removed from the itinery.
Le Harve - yes, again extra €€. I decided to stay on ship. Apparently walking was easy though if you don't mind a 20-30min walk.
Prices of shuttles were around €10-15pp. I thought it was a rip off. But plenty were taking them.
Seemed to be whatever you fancied. On night was "elegant" and there was the odd ballgown. But most were just dress/shirt. I had nice dresses for the evenings (cocktails mostly) and felt absolutely comfortable. But others in jeans, t-shirts were equally so.
Every night was a different themed party with a dress suggestion, but the only one ppl seemed to go in for (except elegant) was the White party. I was wholly bemused by this. How people owned so much white and then chose to bring that highly impractical colour on a cruise! I didn't know it was a thing. Laughed my head off when the chef suggested meal that night was tagelllatelli - I wonder how many ppl were less white by the end of dinner! Anyway, lots wore white, but equally (like me) many didn't. Took great fun wearing black and walking through the party!
Enjoy your cruise of the industrial ports of Europe. I did. Seriously consider getting the thermal pass if your into that sort of thing. It was seriously good!