#2. On a separate note, why are the white wires to the disposal connected to the line out to the next outlets?
That is completely wrong. I don't know what they were thinking!
It sounds like you have 2 completely separate circuits in this box: One that powers the disposal, and a double circuit (MWBC) that powers the receps.
When you have multiple circuits in a box, neutrals must stay with their partner hots. There are several reasons, but one is that neutrals don't have breakers. They depend on the idea that neutral will carry the same current as its partner hot (or with MWBCs, <= the current of the hots due to how they work).
As things are, that disposal neutral could be returning 2 hots' worth of power, and could easily overload.
I don't know where that "probably to next outlet" leg is going, but it needs to make up its mind which circuit it is on! I would put it on the disposal circuit, because that is less likely to violate Code.
All the outlets have 2 blacks, 2 reds, and 2 whites AND a ground running to them. The specific outlet I want to change is also in the same box as the disposal switch. See picture below.
As far as I understand, you can't have all 7 wires to one GFCI outlet because there's no breaky tab on the white side and the circuits will all mesh and short out.
#1. Can this outlet be turned into 2 GFCI? Or will I just have to get an electrician to figure out wiring 1 breaker to each outlet?
This is called a Multi-wire branch circuit. Those are legal, but you must follow several rules, which "the last guy" ignored.
Rule 1: Neutrals must be pigtailed. That is, neutrals must go to a wire nut and be joined with a short white wire that goes to the recep. Neutral must stay continuous even if the receptacle is removed.
You notice that grounds already work this way. In an MWBC, neutrals must too.
Rule 2: Back at the breaker panel, the breakers must be handle-tied or be 2-pole. Can't be independent breakers.
One very fast way to install GFCI protection is to fit a 2-pole GFCI breaker. Boom done, you don't need to fiddle with anything else.
Now, as far as fitting GFCI protection, you have several options other than "blowing out the box to a 3-gang". Nothing goes on LOAD - leave the warning tape on it. There is no way to use LOAD with an MWBC.
Put one GFCI recep only on one leg - completely ignore the black leg. In that case, the neutral pigtail goes onto the silver LINE screw. The red wires go on the brass LINE screw. The black wires connect to each other, but no longer connect to the socket.
Blow out to a 3-gang box and fit two GFCIs. In this case you get one GFCI per circuit. For simplicity's sake, I say pigtail white neutral to both GFCI's silver LINE screw. One GFCI has red wires on its brass LINE. The other has black wires on its brass LINE.
Fit 2 GFCIs abreast in the 2-gang box... one of the GFCIs is a half-switch. Same as above, except the 2 pigtail wires (which are for the switch) go to the 2 wires going to the switch currently.
None of this will do anything to provide GFCI downline protection to other outlets in the kitchen. If you need to do that, it may be cheaper to use a 2-pole GFCI breaker. You will need to fix the poached-neutral problem, because GFCIs don't tolerate that.