Is this customary marriage, which took place in a cafe in the presence of two witnesses only from the husband's side and without signing any contracts or presenting a dowry, considered a valid marriage in Islam, bearing in mind that the husband is already married and has children, and has not moved in with the wife or spent on her?

The marriage contract is not valid except with the presence of the woman's guardian (wali). If the guardian is not available, he can delegate another person. It is not a condition for the marriage contract to be performed in the mosque, nor for the witnesses to be from the woman's side. Rather, any two just witnesses suffice, along with the public announcement of the marriage. Writing is not a condition for the validity of the contract, but it is important for preserving rights. The husband must pay the dowry (mahr) and maintenance (nafaqa), and his failure to do so does not affect the validity of the contract itself.