Having multiple calendar plugins shouldn’t create these issues unless they use the same custom post type or URI parameter names. Which they probably wouldn’t be but if you wanted to check EveryCal uses types and params starting with ‘ecp1’.
Now onto specific issues:
I’ve only done basic testing on changing the initial date of a repeating event. It works in simple cases however there are plenty of edge cases where it might not work. I expected repeating events would not be changed in this way (mainly because the data shouldn’t need to change). Maybe try changing the date of a one off event – if that works then try to add repeats to that event.
The repeats cache can’t be erased from the WP admin; this is a feature I’m planning to add in the future. You could do it using your database administration tool though: look at the ecp1_repeats table.
I can’t explain why repeats would work on one day but not on another day. Theoretically you should be able to have many repeating events every day of the week. There is an upper bound from the PHP execution time though which would make this impractical for a large number of concurrently repeating events. I have a few ideas though:
a) It might be related to Saturday being the 1st of March. The front end calendar requests all events in the range it needs which somehow might exclude the 1st of March – however this shouldn’t be the case for 2014 where the week starts Monday 24th February.
b) What are you using as the event title, description, etc? Are you using any non-ASCII characters? If you are try removing them as these might cause the event to be dropped if the characters can’t be encoded.