Ah! So a few things:
1. I'm assuming when you test adding a new record it takes you to the new Intersecting Entity you just created, right? Showing only the Name field?
You'll need to customize that form to put on the appropriate fields that you created (look-ups, currency).
Then, you'll need to customize the VIEW that the sub-grid uses on the Lead for this Intersecting Entity, to display the fields (look-ups + currency, etc) on the sub grid itself.
You can certainly remove the Name field from this record as well. You'll have to make it "not required" first.
Again, this will have to occur on the Intersecting Entity record you created (Settings > Customizations > Customize the System > Your Intersecting Entity > Forms > Information > Do your customizations.
2. Since you are creating an Intersecting Entity, you'll have to create the Intersecting Entity record each time you add it to the Lead. This means a new record will open up, like you are experiencing now.
Then, once the record opens, you will have to fill out the Competitor (Account) in the look-up you created, and can look-up Product in the product look-up you created.
First, as per #1, you'll have to add this to the Intersecting Entity form, of course.
Then, once you save the record, it will show up on the sub-grid in the Lead record, displaying the associated Product and associated Competitor that you have filled in. You can create as many of these intersections as you want, allows for multiple Product/Competitor combinations to multiple Lead records.
As you can see, it functions a little different than the in-line search box of the out of box N:N or 1:N relationships. That said, it allows for you to add a more robust intersection, and store data about the intersection.
If you need to add a Competitor (Account) each time that a Product is added to a Lead, this is really the only route you'll have.
There are other options if instead, you were OK with adding a list of Competitors that are associated to the Products (Multiple Competitors to Multiple Products).
If that is OK for your requirement, you could implement an out of box N:N between Account and Product, and associate the Account's with Products as needed.
From there, you could add an out of box N:N between Product and Lead, adding a sub-grid of Products to the Lead record, using the N:N relationship.
The downside of this approach is that to see the related Competitors (accounts), you'd have to click into the Product record itself, and would not be able to see it just from the sub-grid.
Let me know if that makes sense!
