We moved from East Anglia to the South Midlands a couple of months ago and are in temporary accommodation until our new house is ready in a few months time.
With regards to moving, we just got several quotes and cross-referenced quotes against reviews. No issues, they turned up spot on time in the morning and we were all unloaded by later in the afternoon.
The only thing was that several companies wanted to do video surveys. If they didn't want to turn up then I didn't go any further with them.
With regards to temporary accommodation I can't help, we are in my late parents house so living rent free and not tied to a move-out date (the house is being sold but the completion date will fit in with our house move).
As for storage we didn't bother. Although we have lots of stuff (including my garage / workshop tools etc) we have it with us in my parents house.
Several reasons for this. Storage is not cheap. There is plenty of room to store it all in my parents house (two bedrooms, dining room, conservatory, garage & sheds are all full of furniture & boxes). If we suddenly decide we MUST have something we just sort through the (carefully labelled) boxes and bingo, there it is.
There are a couple of threads running about using removal firms to do your packing.
If (like us) you are leaving most of your possessions boxed for a few months then do it yourself using your own boxes, no point paying twice or renting boxes when there are plenty of good options out there (I have bought lots of twin-wall cardboard boxes & plastic Tote stacking crates and when I have finished with them, those I no-longer want to keep will be sold or passed on to other movers).
With regards to moving for the second time, this past week we have had more surveys (once again those requesting video calls weren't followed up) so when next month we get a date I already have the quotes ready & waiting (by the way, all removers have been VERY impressed with the quality of the boxes & packing, the horror stories they tell show how a lot of people DIY haven't got a clue).