The five mistakes in the email are:
1. 'Lets go' should be 'Let's go'
2. 'Why not seeing' should be 'Why not see'
3. 'Shall we to go' should be 'Shall we go'
4. 'Why doesn't we meet' should be 'Why don't we meet'
5. 'Why not to call me' should be 'Why not call me'