Fpga timing/timing simulation

Hello Everyone,
I have got a bunch of questions for the fpga experts out here.

I am trying to create a design in fpga using softcore MAC controller and MCB on spartan6 device. All the timing constraints are met in post-place and route but still i see the firmware crashing some times.

I didnot perform timing simulation? Is it necessary? .How is it different from timing analysis? What are the defects that can be found out using this?

Even though the timing constraints are met the worst case slack as reported in the post -place and route report for some of the constraints is very small. (like the timing is met by + 0.0005 ns ). Can there still be timing errors even there is no negative slack? If so, What is the minimum value of worst case slack that is guaranteed to work in hardware ? Should i tighten the constraints further? ..overconstrain the design a little bit?

Is there a way to check that all the path in the critical path have been constrained?

and lastly,
I got a pushbutton signal coming into the fpga ,(on the critical path), I registered it(125 Mhz) , passed through the combination circuit ,then used a synchronizer(2 d flipflops(30 Mhz)) to cross the clock domain ( from 125 Mhz to 30Mhz) .Is this a correct way to synchronize a asynchronous signal?

Would really appreciate if some one can help me solve the problem.

1. About the timing issue: some "first order" questions:
a. Is the speed grade defined in the project match the actual part number stamped on the device?
b. Are all operating supplies in the voltage range spec?
c. Operating temperature (does the device get too hot)?

2. About the pushbutton: synchronization method sounds OK. Did you consider de-bouncing the pushbutton signal before using it? (pushbutton / switch tend to bounce - few random pulses during the first few mS of the transition, that upon application, may need to be (digitally) filtered).

FPGA timing constraints must be carefull, you should better put the clocks in the xilinx clock primitives, such as BUFG, BUFGMUX etc. this could be help.