Form
EEA3 is the correct form. You can only apply by post.
PR status is acquired automatically after exercising treaty rights for 5 years. The HO just confirms this status. You don't need to apply for it but you can if you wish.
In my opinion, having PR confirmation is mainly useful in the following cases:
1. Applying later for British Citizenship in cases in which your 5 years of exercising treaty rights is not straight forward (you can apply directly for BC after 6 years. You don't need to have PR confirmation but if your case is not straight forward, better to make EEA3 application first as it is free).
2. Applying in future for British passport for children born in the UK - Children born in the UK after obtaining PR status are British from birth and can apply for a British passport. Having PR confirmation will make the passport application simpler (no need to provide 5 years treaty rights evidence).
3. Supporting EEA4 (non-EEA) application as you are sending your documents anyway.
For EEA children, you will need to provide their passport + birth certificate (as proof of relationship to you). I would not bother with the children application as it adds no value to them (if you wish to apply for BC later, then it is your status that matters, not theirs).
