It is not permissible to communicate with a woman, other than one’s wife, in such a way that arouses sexual desire, even if it was through the internet.

Yes, it is permissible to do so provided that it does not lead to a prohibited act, like touching or looking at each other with lust and the like.

This is permissible.

It is permissible for the man to talk to a non-Mahram woman, but this might encourage him to commit sins, like looking at her with lust and sexual arousal. Spending long periods of time together could lead to social problems, like making people talking about your relationship or the parents of the girl not being pleased with this. For these reasons, we advise you to correspond with her parents in giving you the chance to get to know her in order to avoid falling into such mistakes.

This is permissible in and of itself, but one should beware from committing sins when being there.

Joking in itself is permissible, but it breaks the bounds of modesty and causes seduction and corruption in most cases.

Going out with your female colleagues is permissible in itself, but communicating with non-Mahram woman causes seduction and corruption in most cases.

Accepting the invitation in this particular case is permissible, but mixing with the opposite sex and the absence of the boundaries of modesty between the two parties might have negative effects and social dangers that leads to damaging the family ties between husband and wife, so observing the precaution from this issue is the wise choice.