It will not affect your credit score by supporting children or alimony. But if you fall behind on your payments and a collection agency involves your credit score will drop. You may even be arrested and sued for the payments you missed as a result.
Mobile phone payment and utility
Many utility service and mobile phone providers will check your credit score before giving you service. However these business providers do not send your payment information to credit bureaus as insurance companies. The timely payments on your utility or mobile phone bills will not affect your credit score at all. Anyway do not let your account becomes past due, because it may hurt your credit score when it is passed on to a collection agency.
Since the rental payment line does not affect FICO score, it will not help your credit score by paying your rent on time. But your credit score will drop if you fall behind on your rent, which will affect your ability to rent or get credit cards and loans.
Checking your credit report
Whenever you want to check your credit report you can go ahead and it will not hurt your credit score. The credit bureaus, FICO, or a legitimate third-party will allow your doing this all the time. Your credit score may be hurt if you let a lender check your credit score.