Does HM have legal custody of all children in the immediate line of succession? In the top 10, for example.?

By British law yes Her Majesty the Queen does have legal custody of her grandchildren, This includes Prince William's children and Prince Harry's children. The anonymous poster is incorrect. The law that states the reigning Monarch has custody of the minor children goes back to George the 1st. The 300 year old law has never been changed, and a European court can't overrule British law.

No in the UK the Grandmother has No legal Rights to Grandchildren if she wants to adopt she gets in Line

She does not. There is no law that grants her custody so she has no legal right. 

Any attempt could be overturned at the European Court of Human Rights (a decree is not the same as an Act of Parliament). The UK's Human Rights Act requires UK courts, including the Supreme Court, to "take account" of decisions of the ECHR. 

As you obviously must know, with the soon to be arrival of the next Sussex baby, we will be going into clouded waters - never been there before.   So yes, HM does have legal custody of royal babies (not sure how far down the line that goes) but how this is going to apply to Sussex child #2 = unknown.   HM is hardly likely to be literally involved in the custody of any of the royal babies after all.

No and no.

See below:

In theory, the Queen can be seen as having custody, but in practice, children stay with their parents, wherever that may be.