Does an induction cooktop attract or repel the base of a pan? My first guess would be that the frying pan would be attracted to the induction coil just as any ferrous metal to a solenoid.  But I recently found out that an induction coil can actually be designed to levitate (repel) the work piece in some designs (induction levitation).
Are the forces between an induction coil and a metal object always attractive/repulsive/neither or is it a function of the induction coils oscillation frequency or something else?   
 A: An induction cooktop repels the base of the pan. 

My first guess would be that the frying pan would be attracted to the induction coil just as any ferrous metal to a solenoid.

If we were to supply a direct current (DC) to the induction coil in an induction cooker, as you say, the coil acts like an electromagnet with fixed polarities and would attract the pan. However using a direct current will not lead to induced currents in the pan. And hence, no heating = no dinner!
Here the case is different, we supply an alternating current (AC) to the induction coil. An alternating current produces an alternating magnetic field. And a time varying magnetic field induces current in the base of the pan as per Faraday's law of electromagnetic induction. These induced currents, known as Eddy currents, cause heating of the pan which then cooks your food.
The following video by Veritasium demonstrates electromagnetic levitation of an aluminium plate of mass $1~\rm kg$. An alternating current is applied to the induction coil with an rms current of $800~\rm A$ and frequency $900~\rm Hz$.
Levitating Barbecue! Electromagnetic Induction - YouTube

Are the forces between an induction coil and a metal object always attractive/repulsive/neither or is it a function of the induction coils oscillation frequency or something else?

The force between an induction coil and the pan is always repulsive. As per the Lenz's law, the direction of induced current in the pan is in such a way that it opposes the change in magnetic field. If the magnetic field weakens, the induced current reinforces the field. On the other hand, if the magnetic strengthens, the induced current tries to weaken the field. The magnetic interactions between the induction coil and the pan is hence always repulsive as their magnetic moments always oppose each other. Somewhat similar to keeping the like poles of two bar magnets closer to each other.
The nature (attraction/repulsion) is invariant of the frequency, voltage, and other parameters as these laws of electromagnetism must hold good. However, these parameters might have impact on the magnitude of the repulsive force. This is the reason why we don't see the pan levitating on top of the induction cooker as it did in the video linked above!
The following quote is from the website - Explain that stuff:

Although your home power supply alternates at about $50–60~\rm Hz$ ($50–60$ times per second), an induction cooktop boosts this by about $500–1000$ times (typically to $20–40~\rm kHz$). Since that's well above the range most of us can hear, it stops any annoying, audible buzzing. No less importantly, it prevents magnetic forces from shifting the pan around on the cooktop.

This explains why we hear a buzzing noise in the video linked above and not in our induction cookers. I was unable to find a source giving the rms value of current that passes through the induction coil. I can say for sure that the peak current must be much less than $800~\rm A$ (used to lift the aluminium plate). Using a comparatively lower current also helps in minimizing Joule's heating in the induction coil, thus reliving some stress on the exhaust fan.  
