Would there be sufficient rolling stock to lay on a fleet of Eurostar ski trains and still maintain the bread and butter service? Especially as ski trains will have much longer out and back journey times than the regular Paris/Brussels services.
Presently, no. However, if the stock was sweated more on the Brussels/Amsterdam route (i.e. at the moment, the Amsterdam arrival is at 1312 and leaves again 1648, most 1005 arrivals in Brussels form the 1252 departure), I'd say 2-3 trains to Geneva a day would be plausible.
Slightly different tack - would Eurostar be able to start services from The Netherlands and Belgium to France and compete with the infrequent TGV and Thalys longer services? Which could also feed their London network at Lille potentially?
No, as it would imbalance the current crew set-up and would cannibalise the Thalys product. In addition, Lille-Europe only has 4 platforms, requiring reversals and the Lille Eurostar terminal is too small to handle so many passengers, it can currently do about 500 people an hour judging but what I have observed from getting trains on Sundays, when they're hourly towards London in the afternoon.
As to extending Eurostar to Geneva as
@Bald Rick has suggested, on the 14th May 2012 Eurostar Chief Executive Nicolas Petrovic told the thelocal.de that Eurostar would not only expand to Cologne but also to Amsterdam, Lyon, Marseille and Geneva over the next five years.
Now I'm aware that Eurostar have actually started services to Amsterdam, Lyon, Marseille but not yet Cologne nor Geneva yet was meant to by 2017, I hope this is still something that they will be doing.
I strongly, strongly doubt this. Petrovic left just before the Amsterdam launch I believe and looking at his profile and an interview he gave which is on YouTube, he is a clear SNCF man. He worked his way up, left, came back again and I believe he used to run an entire region. If you look at the Directors team on Eurostar's website, there used to be 3 or 4 who were all from SNCF/Keolis etc, now I can only make out one (the COO?). The new CEO used to work for Arriva and Easyjet and they have a new 'Digital MD' who was from Cathay Pacific, BA and Expedia. This would signal that their priorities are to consolidate the current Eurostar offering to an airline model with an emphasis on digital service and NOT the SNCF's "trial and error", railway-led schemes, it has done with its other affiliates. The last time I travelled from St Pancras, there was a new hair salon in the departures area, there seemed to be a mini-duty-free shop where the WHSmith was and this was despite clear operational issues causing delays and misinformation - the priority seems not to be the railway, but how can we squeeze money out of customers through extras on the journey and 'package it as a travel experience'. Fair enough, not many rail companies have executives with heavy railway experience, but you can feel the change in Eurostar direction. Go on the website and try and find the timetable for instance - virtually impossible. Try and find their service information - not clear, but doable. Try and book a hotel or get 'inspiration for your romantic city break' - MAIS OUI!
For now, I think they will continue to work on sorting out Amsterdam, and they probably will tinker with the Marseille train - its days of operation seem to vary every season and perhaps, only perhaps we might see a Bordeaux (they'll keep an eye on how Thalys does). But as for Geneva, Cologne or Frankfurt - no. I'd even speculate them killing off certain services - reducing Ashford calls (again?!), simply cutting the number of trains on Sundays to avoid Sunday overcrowding (perhaps running an extra Lille service to compensate?), converting one/two Paris trains a day to 'OUIGOesque' standard class only, using the Izy route via Longueau and using the Paris Nord redevelopment to turn it into an airport terminal.
Also you could have services which partly empty at Brussel and then fill up for Keulen (Dutch spelling). In other words, genuine competition, as envisaged originally.
This doesn't work on the way back though, because passengers need to be fully checked/cleared to go through the Channel Tunnel to London. The only other solution would be to do it how the Brussels>Lille service works, by boarding all the London passengers separately first, then allowing the non-London passengers to be escorted by security officers to the back of the train which is locked off to the London passengers and guarded until the last stop before the Tunnel. The issue with this is that it means you cannot sell seats as efficiently as you could otherwise, it increases dwell time at train stations and requires an adapted terminal such as Brussels, requiring even more space and requires extra security. It's highly undesirable. The current plans for Amsterdam>London services is similar to this and still hasn't been achieved despite the fact that trains are already running, there are crews based in Amsterdam and NS agree with the service running- so as for Germany, chances are very slim.
A better solution would be reviewing timetabling to get connections at Brussels to the most attractive as possible and for DB and E* to consolidate their pricing to offer a better 'London Spezial' type ticket, with clear rules and a business class variant too. If they managed to get ICEs to link up nicely with the twice-daily Brussels-London non-stop Eurostars, Cologne through-trips would be very attractive.
I've said elsewhere that what I think Eurostar needs to develop is more direct high speed lines.
Specifically, a Y-shaped line starting at Antwerp, bypassing Brussels to the west, with a stop at Ghent after which there'd be a branch to Calais and a branch to the existing triangular junction to the east of Lille. That could get London-Amsterdam trains down to 3 hours and allow more capacity through Brussels.
Secondly, perhaps more ambitiously, a base tunnel under the Sonian Forest would allow Brussels to be bypassed to the South-East as well, allowing more direct London-Cologne trains.
That will never happen. Eurostar do not own/operate any of their infrastructure apart from their Stratford depot and the route you propose would involve ripping up many, many WW1/WW2 graveyards and important conservation areas across the Flanders region meaning it would gain almost zero support. There is already the Lille-Antwerp line which offers a good service but trains are only hourly, have in excess of 10 min dwell times at some stations and attach/divide portions en-route. If E*'s Any Belgian Station tickets allowed you to travel via Lille as well as Brussels, so you could travel to Bruges or Ghent via Tourcoing and Courtrai, and the Lille-Antwerp trains did not have portion working or long dwells, then I'm sure the service would be better used and become an attractive offering (although the current Eurostar I believe provides good journey times and the right capacity, although there should be 1 or 2 more trains on Fridays and Sunday/Mondays, with 1 or 2 more Lille stops).