The Biot-Savart law for calculating magnetic fields due to a current is presented in my freshman physics book as a general way of getting B from I. But there's no time delay implied by the integral. Can I just manually throw the time delay into the integral? For instance, to numerically calculate the magnetic field at some point due to a time harmonic current in, say, an antenna, can I just: 1. Break the antenna up into known discrete current elements, 2. Calculate the delay between each element and the point of interest, 3. Use Biot-Savart to find B due to each element, and multiply each term by it's associated phase shift (exp(-i*w*delay)). 4. Add em up. It may not be computationally efficient but it would be simple. But would it give the right answer?
Part of the Biot-Savart derivation sets all the time derivatives in Maxwell's equations to zero (current is and always has been circulating in the loop), so I'm not sure how one would be able to use that result when there is time variation. You could perhaps calculate the vector potentials with the retarded time current integrals and then take the curl. That's probably very messy so maybe there's a better way?
Pure magnetic fields to not radiate, nor do pure electric fields. In a half-wave antenna, the input impedance is about 72 ohms, and the current is maximum in the center, and the voltage is maximum at the ends. You first should study both transmission line theory and antenna theory and design. Signal velocities in transmission lines vary from about 0.66 c to 0.98c. When you have an open-end on a transmission line, using a sinusoidal excitation, you get a standing wave similar to the voltage and current distribution on an antenna, except without the radiation reaction. The output radiated output signal is electromagnetic radiation, with both the electric E and magnetic H fields perpendicular to the direction. The ratio of E/H = 377 ohms, the impedance of free space.
The Biot-Savart law applies only in magnetostatics, i.e. when B and I do not vary with time. If you read your book carefully, you'll probably find a statement to this effect somewhere. (It may be easy to overlook, depending on how well the book is written.) If the current varies "slowly" we can often use a "magnetostatic approximation" in which we ignore effects caused by changing B fields inducing changing E fields which in turn induce more changing B fields etc. At a high enough frequency (how high depends on the situation) this approximation breaks down.
If you want to calculate the radiation from a time harmonic field on a wire, then you are better off looking at an antenna textbook for rough approximations and full wave solvers for complex antennas and more accurate results. A method of moment's book like Harrington's or Walton Gibson would be good but Balanis' books would also have the material.