If you get a P45 from DWP in time, give them that, if not, complete the new starters form. It won't make one iota of difference as your first pay date will be the first pay date of the new tax year, so previous earnings/tax paid will not matter.
Therer are huge employers (Civil Service for instance) who have cut off dates for payroll information very close to the beginning of the month, so can't process the information in time for the end of the first month. Usually you would get an 'advance' at the end of the first month and then 2 months' on your May payslip. with 'recovery of the advance' on there, so you get the money on time and the paperwork catches up later. If this is an ordinaruy size employer you should get paid at the end of April.
You are within your rights to raise flexible working at any time, as you are within your rughts to raise anything at any time. If you mean 'is it a good idea' only you can judge that. You cannot know, until you have settled into the role, whether or how flexible working will work for your employer. Once you are settled it may be appropriate to rise the issue, with an explanation of how it would work and how any potentil negatives for the employer may be overcome.
