Does the ownership of money saved for children transfer to them merely by placing it in an envelope designated for them? Is it permissible to dispose of it for purposes other than their affairs? Should it be zakat-ed with the father's money or separately, knowing that their ages are under four?

If you intended the gift of money to your children for their future benefit in their studies, then they owned it as soon as it was set aside, and zakat (definition) became obligatory on it if it reached the nisab (definition). Your disposal of it for purposes other than theirs would be a revocation of the gift, which is permissible for the father. However, if you only intended to keep it to spend it on them when needed, then the money belongs to you, and its zakat is on you.