Ok, looks like you need to start taking charge of your fertility a bit. Research for yourself what each procedure is and the terminology he uses.
The dye test is essentially a secondary test once you've had the initial tests and has nothing to do with fibroids, generally. Many many people have this test whether or not fibroids are present. It's either an hsg or hycosy and it checks to see if there are blockages in your Fallopian tubes that prevent sperm meeting egg. It is also used to see if there is excess fluid in the Fallopian tubes (called a hydrosalpinx) which makes ectopic pregnancy slightly more likely and creates a hostile environment for fertilisation. Btw, I haven't had either test, but have researched extensively myself so that I knew what the doctors were on about, what my possible next steps were, etc...
10cm is a reasonably large fibroid. It's impact on fertility will depend on its exact location. There are things that they can do to try to shrink it (by cutting off its blood supply in some cases) or remove it, but removal creates both scar tissue and a weakness in the uterine wall, which could increase chances of rupture in pregnancy.
Thais is your fertility, your future. The doctors are there to help and give opinions but you need to be as informed as you can do you can make the choices that affect you and your ability to have children.
We are obviously happy to signpost and provide as much info as we can on this board, but ultimately the decisions are all yours to make, as you are the most invested in their outcomes.
It's a minefield. Good luck.
Might I suggest some books? 'Taking charge of your fertility' is a good place to start.