Hello you need a roofer to come out and inspect the chimney it could be the lead work that’s needs replacing or it could just need a chimney cowling put in the chimney pots to stop rain water coming down them but no The chimney has not got to come down hope this helps

Hi as long as its supported correctly then no it doesn't,although if the chimney stack is redundant then from a leak point of view then yes you would be better off removing it but providing the flashing work is all good and the chimney capped off then you should have no issues at all
Your problem sounds like the chimneys not capped off and fitting a cowl depending on what is on top of your stack to what kind of cowl you will need should stop any water getting down the stack at the same time allowing an air flow.hope this is of some help to you and best of luck with solving your unwanted water feature