to try to influence policy
The only reason anyone files a brief as a "friend of the court" is because they have an interest in the outcome. While policy is generally made by the legislature, with the approval of the executive branch and that is usually where lobbyists work, the judicial branch's power to declare a law constitutional or un-constitutional gives them power over policy. So when a policy that the group that pay the lobbyist have an interest in the outcome of comes before the judicial branch on appeal, they might very likely file an amicus curiae brief.
to have a voice in the government
To gain more knowledge on the subject of interest - and share their knowledge with others.
environmental
So the president can reand and sign it to be a bill
There are many thing the court looks at. The court will see what is in the best interest of the children. If you stayed home with them as married you might not have that possibility when divorced.
It depends on the size and financial abilities of the group. A well-capitalized group may register as an 'interest group' or 'lobbyist' who actively try to influence American laws and policy, whereas a local group of concerned citizens may only be able to vote a mayor or school board member in or out of office.
So the president can reand and sign it to be a bill
The non-custodial parent can petition the court for such an arrangement but it will be up to the court to decide. An argument could be made that might not be in the best interest of the child if he/she won't be able to maintain the same school and group of friends. You should consult with an attorney who could review your situation and explain your options.
Without knowing anything at all about the case in question, it is difficult, and could be dangerous to try to decipher brief notations and scribblings on legal or court documents. It MIGHT mean - that the outcome of the case is known to the parties involved but will held until it is announced in open court. Or - - it might not.Ask your attorney, or ask the Court Clerk's Office what it might mean to them.
make U.S. policies favorable to a foreign government
I'm not sure what you mean by 'intervene'. If you mean, become a party to the case, you would have to have some reason to do so. For example, if two people were suing over ownership of a piece of property and you figure out it's really yours, then you might have grounds to intervene. The court would have to decide if you have standing to do so. Now, if by 'intervene' you mean submit a friend of the court brief, you have to ask the court for permission, and there may be restrictions, e.g. to ask the U.S Supreme Court for permission to submit a third-party brief you have to be an attorney licensed to practice and a member of the bar of the Supreme Court.