Can I use a power strip with max 250v to a 120v power outlet without a converter?
You can use your EU power strip in the US with a plug converter.  

In general, applying a higher voltage than an electronic component's rating is bad; applying a lower voltage shouldn't hurt.  The limitation is on current handling.  Say you have a 2400W load.  At 240V, you'd draw 10A from the power strip.  At 120V, you'd try to draw 20A, which may cause a circuit breaker to open-circuit, hopefully before it damages the power strip.