Heading to Kenya in one week. Thinking of bringing older Blackberry Curve to use there. Anyone tell me if I would be able to get a local SIM card for this phone?
Shouldn't have a problem, Blackberry have a business presence in Kenya, and the major networks support Blackberry services and sell Blackberry smartphones.
Safaricom is the the network with widest and most consistent coverage in Kenya, though other networks can offer cheaper International calls

Correct me if I am wrong, but I believe you will struggle to get a SIM without producing valid identification, especially when buying 'over-the-counter' SIM cards
You are correct that you have to complete a form and to show your passport (or your ID card if you are a Kenyan citizen) when buying a sim as all sims are now registered. Go into one of the networks' shops, they usually complete the form for you, show them your passport (it makes it a little simpler if you have a photocopy of your passport to give them). Only takes a couple of minutes, no struggle involved. There are Safaricom and Airtel stalls in the current, temporary, Arrivals Hall at JKIA Nairobi selling sim cards, credit, etc
(Are there any other kinds of sim other than an 'over the counter' ones?)