Seasonality is all about what the weather is like when you will be climbing. You can walk in most weathers but climbing on Island Peak is restricted by snowfall (avalanche danger) and/or high winds (cold/being blown off the ridge). The Spring season is pre-monsoon. That means that the weather gets warmer as your trip goes on and that the end of your trip is the more likely to be affected by the monsoon. As the climbing takes place at the end of the trip, it is the more likely to have problems. In Autumn things are reversed. Your walk in may be affected by a late monsoon but the climbing will probably not be. However it gets colder and you will need better equipment to climb in the cold. In practice that probably means buying or hiring a good down jacket and decent windproof trousers.
My preference is for Autumn. Once any lingering monsoon is out of the way, the weather tends to be quite predictable. In general the sky is clearer and views are more interesting and photogenic.
However you should bear in mind that high mountain weather is unpredictable. You should be prepared for disappointment. I have a success rate of 60% over twenty trips to Nepal!
I prefer to stay in tea houses/lodges on the walk in. IMO it is more comfortable and probably causes fewer environmental problems. It also spreads the money around the locals a bit more. It is a matter of taste.
Both companies are well known and should be reliable.
