This is the universal conundrum of parallel vs. serial connection of cells/batteries, for both discharge, & re-charge. There are pros & cons for each scenario - efficiency/losses (P=I^2.R), safety (Lithium is an unforgiving mistress), convenience / accessibility, longevity, equalisation.
RC hobbyists trade convenience with a willingness to disconnect packs from their gear, and then connect them to a charger, which gives them flexibility to arrange cells in whatever series or parallel arrangement makes best sense for each discharge & recharge phase of their play-flow (work-flow - geddit? ;).
As you say, your application doesn't facilitate this physical flexibility - the cells aren't easily accessible, nor is changing harnesses. You could engineer a complicated switching scheme with either relays or mosfets, to allow you to reconfigure the array for discharge & recharge, but that's a lot of work, a lot of cost, and extra copper/mosfet losses.
If it were me, I'd suck it up & go for a Li-Po recharge solution that supports the number of series strings with cell/pack-balancing capability (which of course is critical in series configurations). That may mean you need to take your power source & use a switching boost voltage regulator to get the voltage you need (i.e. more than the number of series cells * 4.2V) to achieve this, and then a charger to match.
If you really want to stick with hobbyist-level chargers like the one in your photo that can handle a handful of cells in series, then one approach I can think of is to power each of those rechargers via isolated DC-to-DC converters (in turn powered from your solar/MPPT supply, or bulk mains-powered PSU) - that lets each charger handle a handful of cells in series, and provides the isolation needed between each charger as the string voltage escalates along the string. Here's some examples of the kind of DCDC modules I mean. One challenge of this approach is how to control each of those rechargers electronically (instead of humanely) whilst maintaining the isolation needed between them, but that's a separate question :).
If you place multiple cells of different capacities in parallel, they act as a single cell with the combined capacity of the two. You should never put cells with significantly different capacities in series, since the lowest capacity cell would discharge before the others. This is also why you should (unlike what the picture you posted portrays) connect cells in parallel when they are not in series, as it helps even out the different cell capacities.
By connecting the 1152 mAh and 871 mAh cells in parallel you would get the equivalent of a 2023 mAh cell, while connecting the 1121 mAh and 931 mAh cells in parallel would yield the equivalent of a single 2052 mAh cell. You could then put these two parallel groups in series, to yield a slightly imbalanced 7.4V 2023 mAh pack.
You should also aquire a 2S balance plug extension wire, cut the male plug off and connect the wires to each parallel group of cells in the pack.
simulate this circuit – Schematic created using CircuitLab
That said, I wouldn't try to salvage old, end-of-life cells from some old laptop when 4 brand new 2000 mAh 18650s can be bought online for less than ten bucks. Just don't fall for the "6000 mAh UltraFire" scams, research online which cells are actually good.
Best Answer
Short answer: do not play matchmaker with Li-ion batteries.
1) As explained in the comments, parallel batteries with different voltages and low resistances will probably catch fire and certainly destroy one another if connected for any length of time. Do not do this. You will cause a fire.
2) Batteries at the same voltage with different capacities will have to remain at the same voltage all the time because they are in parallel. In a perfect world, they would drain according to their capacity, remaining at equal voltages the entire time. This is not a perfect world. Because of things like internal resistance and general errors (both age and manufacturing defects can mess with the batteries specs), there is a high chance of overheating. I would strongly advise against trying this. If you were really desperate, I would use diodes on the terminals of each battery to make sure that the current is always flowing in the correct direction (away from the batteries). The voltage across the diodes will increase if the batteries voltages diverge, but the diode will (hopefully) be able to handle it. I would choose diodes rated to at least half the voltage of the largest battery just to be safe.
3) Batteries in series are, actually, largely fine. The voltages will add. In terms of which ones would drain first, it would be a bit complicated, but you should stop using them after the first battery in the chain is drained. That means that if you have a 100 ah and a 200 ah in series, you would only get 100 ah total. Once the first battery is drained, the internal resistance will rise and it will start heating up. This can result in cell breaches and the like. I would strongly advise against using them after this point.
If you have any problems with any of the batteries, dispose of them safely. They are very dangerous and not something to be toyed with lightly. For god's sake don't touch the terminals together either or you will wreck them permanently.