Tier 1 General dependent visa application

[Sam Kaz]

Hi,
My HSMP extension is due in May 2013, I have 3 dependents (Husband and 2 UK born children). My husband and both children have HSMP dependent visas on their passport.

My ILR is due in December 2013 on basses of 10 years long residency.

Due to increase in Fee from April 2013, I am planning to apply extension for me and my husband only. We don't have any plans to travel abroad.

QUESTION:
My question is about my children’s visa renewal application, I heard if child born in UK and he is not travelling than he doesn't need visa to stay/remain in UK and for naturalization, they need only Birth Certificates and parents’ passport having ILR on it. They don’t need their own Passport with valid visas. And I have been told by some friends that even though your children already have dependent visas but it doesn't matter whether or not you apply their extensions.

If I don't apply for children renewal;
will they be consider illegal?
can my extension application be rejected as I am not applying for them?
Without having valid visas on children's passport can affect their naturalization process?

Looking forward to quick and accurate response.
Thanking you in an advance.

Sam