Solar wiring, chase through to roof?

The cell interconnect tabs run from the front of one cell to the back of the next, which is what creates the series string of cells. Depending on what you have they are usually solder coated copper ribbon. Google "solar cell tabbing ribbon", it is available on ebay if you want more. 

The tabs can be a real PITA to work with especially on the cell fronts because they are machine soldered to the cell metallization layer, which is very thin. Try removing the tabs from the backs of the cells rather than the fronts, there is more metallization on the cell backs to re-solder to.  There should be visible spots where the tabbing machine attached the ribbons to the cells, if you place your soldering iron on those spots while gently pulling the tabs they should come off, one spot at a time. 

ETFE (Tefzel) is the front superstrate material in most "flexible" modules.  The encapsulant is usually EVA (ethylene vinyl acetate) in any module. In the layup there is the superstrate, an EVA sheet, the cell circuit, another EVA sheet, then the backsheet, usually PVF (Tedlar). The layup is placed in a laminator which de-airs the assembly, then cures the EVA under heat and pressure. Once heated properly long enough the EVA becomes highly cross linked so a really tough material. IOW, you're not getting that module apart without destroying the cells. If you want to try anything at all try a heat gun. If the module wasn't properly laminated and the EVA cross linked to begin with it remains soft and flexible and will peel off.  It will drip and burn though if you get it hot enough.

There won't be both series and parallel connections between the same cells, that would just short them out, so I'm curious what module you're working on.  Some module designs use separate conductors concealed behind the cells to return the cell row output back to one end of the module, but usually there is a bus bar that connects one row to the next which is reversed so the series string can just continue on back and forth along the module. They're not Sunpower cells with all the contracts on the back are they? 

If you want to make your own custom modules buy a stack of cells and some interconnect ribbon and bus bar material. Its really pretty cheap nowadays. That part is pretty straightforward DIY, fun to come up with a design that fits where you want it to go. and has the voltage you want.  You just need a good soldering station and some jigs and fixtures.   Its the encapsulation that's hard to do in your garage, unless you want to build a laminator, you'd need a vacuum pump and bag and silicone heating blanket. Or you could go for poured urethane. 

None of this will save any money over just buying off the shelf of course.