You are a citizen of the European Union and as such entitled to freedom of movement. Authorities might be concerned whether you are running away but a simple letter from your parents will convince them you are not. You will find the Documents for minors page useful.
In addition to their own valid travel document (passport or ID card), ...
I suppose you're flying Air New Zealand's London-LAX-Auckland flight. Then their site says you do need a visa or ESTA:
Changes to LAX transit process for customers travelling between
Auckland and London Heathrow on NZ1 and NZ2
On arrival at LAX, transit customers continuing on the same ...
If this is for a trip in 3 months and you’ve already been denied, I would do my best to try to get either a refund, reschedule your trip for a year in the future, or change to another destination country. In other words, you’re not coming to the USA this year.
There’s no appeal process for the current application (except for situations where there was a ...
If you are Croatian citizens then yes, you can do this. A valid national ID card is sufficient for any EU citizen to enter any other EU country.
This is specified in directive 2004/38/EC. Article 5(1) says:
Right of entry
Without prejudice to the provisions on travel documents applicable to national border controls, Member States ...
I travelled quite a lot around Europe without adults at that age, but I mostly was with my sister. The biggest practical difficulty when you're travelling on your own, is looking after your possessions, especially (for example) when you're asleep or taking a toilet break. The best way to reduce the risk is to travel light! Remember that other people are ...
You will need a Visitor Visa. The linked page describes all the steps you need to apply, and what information to submit in your application.
Unfortunately Croatian citizens are not eligible for the Visa Waiver Program (VWP), which makes this process a lot easier for citizens of certain countries.
From what you have described this is no problem. You may need to explain the situation if you approach US CBP together, but you are doing nothing wrong. It is perfectly allowed to do tourist activities on a business trip, and two people travelling together do not need to be travelling for the same purpose.
Since you are supporting your cousin financially on ...