For most of day-to-day life, Euro cash and a credit card is fine. You need a credit card with a chip to buy train tickets, but it doesn't need to be a French one. Many utility bills can be paid by credit card.
On the other hand, some subscription services want to have a debit authorization (“autorisation de prélèvement”) on your account. This requires an account with a French bank (some services may not accept a foreign bank). Debit authorizations are bad because they allow the payee to take the money with no limit on the amount; unfortunately, while you can refuse this system, some companies charge extra for not using it. ISPs and phone service providers are especially bad in this regard.
Checks were traditionally heavily used in France, but they are on their way out. Nowadays, there is only one area where the use of checks remains strong, and not being able to pay with a check on a French bank may be a problem: paying for housing — rent and condominium bills (“charges”). If you don't have a checkbook, you'll still be able to find housing, but possibly with extra hassle.
Another area where checks are occasionally used is for guarantees — “chèques de caution”. This consists of you giving a check which is only be cashed in case something goes wrong: for example, you rent something, and the check will be cashed only if you don't return the object or if you damage it. The more modern places, such as car rentals, have switched to holds on credit cards instead, but the local power tool lending club might still want a check or cash.
If you're employed in France, then normally the employer will deposit your salary into your bank account. As far as I know your employer has no obligation to accept a foreign account, though I can't think of a good reason to refuse now that there are no extra fees given that your existing account is in Euros.
I recommend opening a bank account in France. Basic services are free, and if you show that you have a foreign account in good standing, a job that earns you money and a stable address in France or abroad, then you're likely to get a checkbook and debit card without any hassle.