How can I have custom fields from lead to show up in the lead conversion window which must be filled in in order to convert the lead?
If you're using Out of the box Lead Conversion then you can not modify Lead Conversion Layout where you add those field.
You can't customize Lead Conversion Layout as Salesforce does not allow to do this, you would have to build a new page using visualforce and apex. If you want to avoid custom solution then you can try below solution:
Decide which fields you want to filled before convering the lead.You can add validation rule on those field
AND( ISconverted=True, ISBLANK(Newly Custom Field) )
The validation rule will make sure that if user to try to convert Lead without having filled up the Field then it will throw an Error message.
When you convert the lead into Account/Contact/Opportunity,do you populate any of the fields on objects if yes then you need to Map the Lead fields with Account/Contact/Opportunity Fields.
Goto Lead-->Fields.Under Lead Custom Fields & Relationships section click on Map Lead Fields and map the fields accordingly. Once you populate the value on Lead,it will populate the value automatically on Account/Contact/Opportunity Custom Fields if you map the fields correctly.