How much bandwidth can I expect from this circuit if i used a faster op amp and another to amplify the output ?
You haven't even stated your requirements yet, so it's a bit premature to look at different op-amps.
The first thing you'll notice bandwidth wise, with that circuit, is the slew rate limitation of the op-amp, which is around 0.5V/µs or 2µs/V, so if your square wave has an amplitude of 20V peak-to-peak, then the rise and fall times will be 40µs. As the frequency is increased, the square wave will start to look more like a triangle wave and when the period becomes 80µs or less (a frequency of >12.5kHz) it will be a triangle wave. Of course, the maximum usable frequency will be much less than 12.5kHz, as by then it's no longer a square wave.
http://www.ti.com/product/LM324A faster op-amp will give much faster rise and fall times. For example the TL072 has a slew rate of 13V/µs or 77ns/V. With a 20Vp-p square wave the rise and fall times will be 1.54µs.
http://www.ti.com/product/TL072?keyMatch=TL072&tisearch=Search-EN-EverythingIdeally the squarewave section should use a comparator IC, such as the LM393 or LM311, which is designed for this, so the squarewave section will no longer be the limiting factor. The op-amp's slew rate will then limit the maximum slope of the triangle wave and thus the frequency, as calculated above.
Another op-amp based amplifier can be used to alter the amplitude and offset.
Which op amp has a null pin and can handle the bandwidth?
There is no need for a null pin. An inverting amplifier with the non-inverting pin connected to the offset voltage will introduce the offset you require.