1) Keep everything diabetes related in your hand luggage, to avoid chance of hold luggage being lost, and also because hold temperatures can go below 4oC and thus damage insulin etc.
2) Yes, do tell security staff that you have diabetes related items in your bag.
3) Best to have a letter, ideally generic so you can use it for future holidays / travel, as well as this one. Many GPs charge for such a letter, so ask a diabetes specialist nurse at your clinic (assuming you have one) instead, as they don't charge.
4) Never been to Lanzarote, and haven't been on a package holiday since I was a child, so can't help there.