The answers to all these questions are pretty straightforward.
Is it considered a desire if I only have a strong feeling of wanting?
No. Your feelings are not a factor. Ever. You can feel a lot of things; but at the end of the day, a lot of feelings are fleshy and have nothing to do with our actual desires. Many are "the desires of the flesh" which are only tangentially related to our actual desires.
Do I need to do anything to make my desire a desire?
People act contrary to their own desires all the time... so, no. People can't always actualize their desires for one reason or another.
If I volunteer to incur all the expenses for my family does this make it a desire when before it wasn't?
No... but it sure goes a long way in demonstrating that it was a desire of yours.
What if I only volunteered to pay for some family members?
again... people can't always actualize their desires. There are many factors to take into consideration. Can some family members afford to pay their own way? Could you afford to pay all; but only paid some? Whose expenses did you pay and why?
Does this affect whether it is a desire or lessen the desire to have the company of my family?
Like I said before, It's a demonstration of your desire.
Does God have to do anything for people in order for His desire to be genuine?
Yeah; but he already did. The reason why I say GOD had to do something- and you don't is because he doesn't have our limitations. He can, and has, and does, and will continue to actualize his desires.
If so, does He have to act equally towards all people for His desire to be real?
Depends on what you mean by equal. He desires that everyone is saved... but not to the point that he will force it on somebody. He'll pay for your ticket... and even come pick you up: but he's not gonna force you to go on vacation with him.