Hello,
For mobile (cell) phone providers, Safaricom have the best coverage and also offer MPESA, which is a mobile banking platform that allows cashless payment for the majority of goods and services. Other providers such as Airtel (Safaricom's main competitor) also have mobile banking, but no where near as good, or well developed.
Internet; I just tend to buy data bundles and tether the phone. 4G is available and prices reasonable. There are also companies such as Faiba, which provide broadband services and may be worth investigating.
Car insurance; Prices tend not to vary that much as insurance is set at a % of the value of the vehicle (assessed at the time of insuring the vehicle). It is something like 8%, with a discount if you have no claims within the period you are insured (calculated on an annual basis). You really have to shop around. Look out for what the policy covers (eg. windscreen cover, stereo cover) and check whether or not these are extras. Cheap often doesn't mean good, as getting a payout could be problematic from a 'budget' company. We used to be with Kenindia - they were very good at paying out following an accident, but certainly aren't the cheapest. Google and shop around would be my suggestion. Bear in mind that for popular makes of car, theft of lights, mirrors and so on are common, often while you are in the car, sitting in traffic - which you will do a lot of, if you drive in Nairobi.
Incidentally, the cost of purchasing a vehicle, in Kenya is high - it would be difficult to get a good family car for less the Ksh 1 million ($10,000). If you plan to bring one with you, check the import rules. You can enter Kenya with a vehicle, duty free for max 6 months, then the vehicle has to be exported, or duty paid. Some exceptions apply, such as foreign military, diplomats, UN staff.
Health insurance: A good idea, as public health care is limited and private healthcare (particularly in Nairobi) is hugely costly, largely driven be greed, rather than a genuine interest in treating you! These days, you need to put down a massive deposit - Ksh 300,000 plus to be admitted to the better private hospitals. Jubilee is meant to be amongst the best health insurance companies. We used to be with AAR, but when they refused to pay out for a procedure (having advised that they would), I cannot recommend them. Most companies have a number of levels of cover - you can go online and contact them to get prices. In 2015 we were paying around Ksh 250,000 per year for cover for two adults and a child. Going for the lowest level of cover may not be the best option, if the unexpected occurs, such as a car accident.
Look out for doctors trying to perform unnecessary procedures, or prescribing costly drugs that you don't really need - both are quite common in Kenya and all about money, playing on people's general lack of health knowledge.