Either party can file any motions it deems necessary and appropriate to advance its case; that includes motions for temporary orders. If both parties file timely motions for temporary orders, the court will hear both sides the same day. (Your lawyer or the clerk of the court will tell you about how to notice the opposing party and other filing requirements.)
Orders on motions in child custody/support cases are always decided upon the same basis: the best interest of the child and who makes the best argument why the child would be better off in their custody. Motions for Temporary Orders should be filed right away (with or soon after the filing of the Complaint for Divorce). Temporary Orders can remain in place throughout the pendency of the litigation and are often converted into permanent orders at disposition. There is no strategic advance in filing first in this instance.
You should discuss these concerns with an knowledgeable family law attorney. Contact my office for a confidential consultation.