Are there any problems caused by setting up my phone before the SIM card arrives? Bought a brand-new Samsung S4 mini and ordered a new SIM card which is due to arrive in a week. So far I've inserted the battery and charged it.

Should I turn it on, even without a SIM-card in it, or would all the settings go awry? For instance, I heard that WhatsApp needs some sort of SIM card (phone number) verification. Will Android keep nagging me about my email address and SIM card etc right from the start until I "give in"?

In short, I can wait till the SIM-card arrives. The question is: Is it better to wait (and if so, why)?
There is nothing wrong in turning on the phone and using it. 
Put the phone in Airplane mode and start using Wi-Fi and install applications etc. 

As for the nagging part, No, Google will not be nagging you to register your device or SIM till you choose to (if you have skipped registering the device in the first screen). Google never registers your SIM, it registers your IMEI/EMSI with your account. So even without SIM you can use Google account.

However, without registering the device with Google, you will not be able to enjoy any of the Google play services. Other services (accounts) can be used without any problem. 

Whatsapp registers your mobile number (not SIM), if you are retaining the number you can activate Whatsapp in this phone (S4 Mini) by receiving message in another phone and entering the code in S4 Mini (the application has to be removed from that old phone since Whatsapp can run only on one phone at a time). 

Also, Whatsapp can be changed from device to device without any issues, just have to re-register it when changing devices for the same number.

In short, just turn on the device and use like normal phone (except for calling). Nothing to be worried about.