If you want to drive your MBP without the inverter, i.e., as you have it set up now, you have to determine what voltage it requires.
If, as is possible and maybe even likely given that it's not working, the computer needs around 20V at the MagSafe input, your setup just won't work because it is outputting 12V. Assuming there are no faulty connections, adding an inverter will make this work as it is. It is a small battery though and won't power your machine for long.
Re the solar panel connections, it's never good to use two different kinds of panels into the same regulator/controller. It probably won't break anything (no guarantees) but it's not optimal. Personally, I would just disconnect the folding one for now. The less interactions in the system the better for fault finding. Maybe hook it up again once you've got it working.
Trying to do this stuff without a multimeter is a bit like trying to build IKEA furniture without an allen key in some ways. It's very difficult to know what's going on if you can't at least measure voltage and continuity. I would also be determining the pinout of the MagSafe connector so I could, amongst other things, measure the output voltage of your AC adaptor. This will tell you straight away what voltage the MBP wants to see.
Once you have a multimeter and you know what pins in the MagSafe connector carry power, you should check your lead for continuity to make sure it's working. The overriding principle with any sort of fault finding is to check everything - never assume something is working or not working, you must test it - and minimise complexity whilst you're working through it.
All in all, it would be much simpler to just use an inverter, as previously suggested. To do what you want, you need one that will put out 60 watts continuously as a minimum. It is hard to get good small inverters, but I can give you an idea if you find something that looks suitable and you can post a link or make and model here, or a photo of the one you have. We use SEA Piccolo inverters for little stuff, but they're pretty expensive relative to what you can get off eBay. The last one we sold was about $450 or so and I see the 12V Shop has them now for $467.