Thank you for this offer!
I think it might be helpful to clarify a few things in order for everyone to understand what you are asking
#Cost of attendance (COA)
The whole cost to attend medical school. COA= tuition+fees+cost of living (COL)+other expected expenses
Different schools count different things as expected expenses. Stingier schools include less in the COA, which reduce the need of the school to provide fin aid for those students, for example, step 1/2 fees, transportation for clinical rotation. Yale include pretty much everything under the sun.
#Unit loan
The portion of the COA not covered by maximum amount of need based scholarship. At Yale the amount is $15k. However, the term is a misnomer because you don't have to borrow this portion if you can finance it some other way. For example, get merit based scholarship, borrow from family or friends.
Now answers to your questions:
1. Theoretically, is there any way that one would receive a financial aid offer to Yale that covers full tuition and COL? Or only full tuition?
Yes. It is possible for you to cover your whole COA by scholarships at Yale if you receive additional merit based aid from whatever source.
2. Is the 15k mandatory(?) loan part of tuition?
See #Unit loan
3. Does Yale Financial Aid (excluding loans) cover any cost of living issues such as room and board, transportation, food, etc?
Yes. The maximum need based scholarship offered by Yale is no less than $80k. 20k more than the tuition and fees. I think the reason for your confusion is that some schools like WashU ( at least in 2015 it was the case) prohibit any scholarship to be applied to COL portion of the COA. Yale is not like that.
4. How does a dual-degree program work financial-aid-wise at Yale?
Med school aid applies when you are working towards your medical degree. Fin aid from other other degree programs covers the rest, which are degree specific.
What you didn't ask but probably was trying to: Does Yale have institutional merit scholarship?
As far as I know. Yale does not have pure merit based aid. There is a special scholarship created to provide additional assistance to exceptional candidates who come from socioeconomically disadvantaged background by paying for their unit loan. The financial aid office has no control over who gets this. It's an independent committee. In the past the amount was quit significant because the unit loan was of greater amount. Not sure what they are going to do now with the unit loan reduction.