I'm an Indian student in the UK on Tier 4 visa, valid until March 2019. I don't get long breaks so I want to travel to various places in the Schengen area over short weekend trips (Friday night - Monday morning, etc), maybe once a month.
The places I would like to visit include: the Black Forest in Germany, Paris (I have a friend here - also an Indian national on a type D visa), the Swiss Alps in the winter, and maybe Norway if I can afford the trip.
Can I explain this in the application to the French embassy if I decide to make my first trip to Paris, and request a multiple-entry visa? Will I require itineraries for all the subsequent trips in my visa application?
If issued, can immigration officers of other countries deny entry on consequent trips because it's a French visa? (I've read here that there are very little border checks between Schengen countries - I'm asking about the case where I'm entering directly from the UK.)
EDIT: An additional constraint is that I'd like to reduce expenses: I would prefer to take cheaper routes into the Schengen area instead of flying to Paris/Munich/Zurich. For example, FlixBus from London to Paris is quite cheap but I will present my passport at Calais, or fly London - Strasbourg to visit the Black Forest because some budget airlines have great deals. This might mean I may not always enter the country/city I plan to visit for the weekend. Could this cause problems?