Is the husband sinful if he does not spend the night with his second wife—with her consent—and visits her daily during the day? Is this considered an infringement on the right of the first wife? And does the first wife have the right to demand a divorce due to non-compliance with an unwritten verbal condition that he would not marry another?

If you agree to waive your right to stay overnight, then the daytime remains yours, and your husband is permitted to enter your premises during the day. He is also allowed to stay with you during the day, but only on your designated day. As for your co-wife's day, he is not permitted to stay with you; rather, he may enter only for a specific need, such as delivering maintenance, visiting the sick, or a general visit. He is permitted to enjoy everything short of intercourse.

Regarding your question about your co-wife's right to seek divorce, she has this right if she stipulated in the marriage contract that he should not marry another woman, whether this condition was written or verbally stated. This is based on the Prophet's (peace be upon him) saying: "The conditions most worthy of fulfillment are those by which you make sexual intimacy lawful."