On the first page it asks date of planned arrival and how long I intend to stay, not really sure how to answer those two.
The date of planned arrival is your chance to decide when you want your visa to start. It will be made valid for 30 days, usually starting 7 days before the date you put... so you want to put the earliest possible date you know you can travel within 30 days of. If this date has passed before the visa is issued, it will just be made valid starting a few days after the decision has been made.
For the length of stay you just put either permanently or indefinitely.
Entering my children's information it asked for visa held and I'm assuming I select 'not required' is this correct? And the two of them will be covered on my visa, they do NOT need an application?
Edited because I checked your past posts:
You need to say what visa your children will require (child dependant of a person settled in the U.K.).
As they are US citizens and not British you will need to apply for a separate visa for each child, plus the financial requirement you need to meet will be higher (£24,800 instead of £18,600) to cover the children as well.
There is no situation where your children will be covered on your visa, they each need to apply for and hold their own visa.
Last question for now, on the last page where it asks for additional information what do I include there? Information about my refusal? Anything in particular in regards to the children?
Most people leave it blank but you can put something there about your refusal if you like, though you should also address the refusal in a cover letter and include the refusal paperwork.
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