Radiation
Need not be a problem, if the planet orbits quiet, settled down star. Note that even in our Solar system, Sun radiation is pretty survivable unless there is a solar flare.So you are looking at K or M stars (you might wait a trillion years or so for the flares to settle down - but on the upside, you'd have very stable star with many trillions of years ahead).
Alternatively, change your "small rocky core" to a "small metallic core" and you can have a full featured magnetic field, making it a non-issue even for Sun-like stars.
Gravity
Let's take Saturn as a model - it's "surface" gravity is $10.44 m s^{-2}$, which is practically the same as Earth. Note that gravity does not depend on the size of the planet alone, but on it's mass (linear) and radius (inverse square).Assuming compatible density distribution (which is not true, especially when under pressure gasses undergo phase change into metallic form, but anyway), mass depends on the volume, which is radius cubed. $r^3$ for mass is cancelled by $r^{-2}$ for the radius and the dependency is nicely linear - for planets with the same density, gravity increases linearly with the radius.
Since the mass is density times volume, gravity depends linearly on density (at the same radius). Therefore, if you increase the (average) density of the planet, you need to scale down the radius by the same amount to keep the gravity the same.
Pressure
No problem, just pick the height you need. Remember the barometric formula. Changing temperature will make it more complicated, but you can still pick up comfortable pressure.
Air
This is where things become interesting. Let's assume (for simplicity) a planet made of oxygen. Taking Saturn as an example, it consists mostly of hydrogen, which has a density $0.09 kg\cdot m^{-3}$ (at 1 bar and "normal" temperature). Oxygen's density is $1.4 g\cdot m^{-3}$ (it's atomic weight (16) times the hydrogen density). Thus in order to keep the gravity at $1g$, recall that the planet's radius should be 16 times smaller than Saturn's - that is $60000 km/16 \approx 3750 km$, i.e. 24
about 60% of Earth.
Now, you do not want to live at 1 bar of pure oxygen atmosphere - it is unhealthy and dangerous - anything combustible will combust rather violenly. You'd need 0.2 bar partial oxygen pressure, which means moving upwards just by 12km or so.
Or you can mix in 75% nitrogen, it will not change the numbers much and make the atmosphere very Earthlike.
Cold
Again, let's take Saturn - at 1 bar pressure, the temperature is 134 K, way too low for unaided human life. But since the planet's gravity and dimension is quite comparable to the Earth, just increase the insolation to Earth level (or slightly more, due to the absence of greenhouse gasses).
To summarize
You have a planet made of oxygen (and some other gasses, like water vapour which you'd rather need for survival), slightly (60%) smaller than Earth, with Earth like insolation. It's perfectly habitable, if you manage to float at the correct level (i.e. using hydrogen or helium balloons). If you drop anything, it will be crushed by the pressure - but long before that, increasing oxygen pressure will make it burn (if combustible) rather explosively. At the pressures at the centre, oxygen will likely turn into metallic form, which is going to create some magnetic field, useful (but not absolutely necessary) to protect from solar flares.
Now the question is, could such a planet occur naturally? The answer is, almost certainly no. You'd need a protoplanetary disc consisting primarily of oxygen (and nitrogen), but no or little hydrogen (which will burn in the oxygen planet, providing much needed water vapour). It is difficult to imagine such circumstances. The planet would have to be deliberately constructed. Also, any life will lack solid materials (such as carbon), since everything will fall down to the core (although, carbon will combust and at least part of it will be recycled in the form of $CO_2$). It would have to be imported and kept floating. 279