Wrong dates in database

Dear users,

When copying data from a website to excel, excel recognizes the dates wrongly.

About half of the dates he sees as: DD/MM/YYYY, while it should be: nothing/MM/DD. For example:
The cell contains the date: 01/04/2029 (The first of April, 2029)
While it should actually be: 04/29 (The 29th of April, no year!)
For all dates it starts with the 1, followed by a month, followed by a year (which consists of the year 2000 + the actual day), which is totally wrong.

About the other half of the cells, it just contains text with the month and the day (which are correct), but excel doesn't recognize this as a day. For example:
The cell contains the text: June 12
When I set the cell format to date (or any other format for that matter), nothing happens, so excel does not see this as a date.

Thanks for your help!