Can you do it? If done as in the diagram, yes. Batteries in series mean that the voltages 'add' and the current is 'averaged'. Batteries in parallel mean that the voltages are 'averaged', and the current's are 'added'.
There are draw-backs. If the batteries are not at least 'nominally' the same size, then the smaller (ones) are going to discharge quicker, which means problems.
The size of the conductors used to do the connecting are determined by the current handling capacity of conductors. A 'safety-margin' in size is very nice, bigger is better, sort of, to some ridiculous point. 'Where' those conductors are doing the conducting means that a smaller size might be possible since the amount of current handled might be smaller.
Can you charge those batteries while they are still connected? Yes/maybe, depends on how 'nominally' the same they are, and just how 'dead' each battery is.
Can you mix different 'kinds' of batteries? The 'quick-n-dirty' answer is, maybe. Just depends on the rate of discharge. 'Simplest' answer is no. The 'complicated' answer is, yes, if done correctly. "KISS" is very good (Keep It Simple, Stupid!) with that sort of thingy... so don't.
- 'Doc