FPGA retriggerable monostable multivibrator?

You just have to make your clock fast enough for the resolution you want. For example a 100MHz clock would give you 10ns resolution - do you really need more than that for your project?

FPGAs like clocks, its what synchronises everything together. While it would be possible to replace (posedge clock) with *, I really wouldn't recommend it as who knows what would happen as it would be very likely that your counter would start developing funny glitches (if it works at all) as each bit of it may have slightly different propagation delays from another.