I think it's unlikely you'll be able to do better than the cheap systems off eBay without spending a huge amount of time. Those systems are relatively rugged, weatherproof and have fairly smart charging systems. Building something equivalent means spending a lot of time finding cheap parts and making the bits you can't buy. If you have a 3D printer that will make it easier, but it will still take time.
I found a ~$40 one that folds and has a 10Wh battery that seemed like the best of the lot on eBay but don't have the link handy. Will edit the link in here tonight.
Given that $40 budget for a DIY system you'll need to start with some scrounged parts. If you can find a LiIon USB battery that will make your life a lot easier - all you need to do is add some scrounged solar panels. But I suggest trying to find all the small solar panels you can to start with. Test them to find which ones still work and what their output is. Work out how to stitch them together to get a single source (both mechanically and electrically).
Then spend most of your money on a smart LiIon charger and voltage regulator to feed 5.1V out the USB port you'll use to get power out of the device. I suggest a battery inside it to buffer the erratic solar input as some smart devices don't like that (my phone, for one). Even a smallish battery will do, a couple of Wh, as long as it can handle the max power you get out of your panels.
I would probably go with either a small 12V SLA or ideally scrounge a complete LiIon booster battery system that has a working charge/protection circuit. Then add a 5V regulated output you're good to go.
but might 36v from a pair of panels damage the actuator circuitry?
So here's the deal. Lead-acid batteries look electrically like a voltage source/sink with a small series resistance, with the voltage level a function of state of charge. 2V/cell (there are 6 cells in series in a 12V battery) is nominal, and if I remember right, their open circuit voltage is something like 1.9V empty, 2.1V full. That covers 90% of their behavior.
Considering that, the "1W@18V" spec of the solar panel isn't going to be able to "win" against the battery, and the solar panel's voltage will be pulled down to battery voltage, delivering probably 0.055A (=1W/18V) at whatever the battery voltage is.
When a battery gets completely full, however, its series resistance goes up dramatically, and the voltage goes up, until there's enough voltage to start electrolysis of the fluid and you get H2 and O2 generation at the terminals and loss of the electrolyte. A lead-acid battery, depending on the type + manufacturer, has a certain recombination rate of H2 + O2 => electrolyte that it can handle; if you electrolyze at a higher current than that, it leads to permanent electrolyte loss (+hence capacity loss)
So there is a safe current that can be delivered to a lead-acid battery continuously, where its own self discharge due to electrolysis balances the charging current. It depends on the manufacture + construction. I wouldn't feel worried about a C/10 or C/20 rate of charge (where C = the current needed to discharge a battery in 1 hour). Garage door batteries are probably > 1Ah capacity so you should be safe with 55mA charging current.
HOWEVER -- I would probably put a (zener diode and resistor in series) in parallel with each battery, the zener diode being about 14V and resistor being maybe 10 ohms or so, so that it keeps the battery terminals from getting charged too far.
Also: if you can, wire each solar panel to each battery (and keep the diodes), rather than the pair of panels in series wired to the batteries in series -- i.e. try to connect the center taps. By doing so, you'll charge each battery independently. Otherwise, what can ruin battery life is if the battery voltages diverge -- the one with the higher voltage will tend to get overcharged, while the other one will tend to get overdischarged and not completely charged.
Best Answer
If they are just bare cells, you probably don't have much chance in salvaging all of them. In my experience of building panels (only a few), I usually end up using 10-20% more panels than actually end up in the panel.
As far as actually building the panel, just solder all of the tabs together in some order (whatever size/shape you are going for), and then mount them between sheets of acrylic. Use some spacers so that the acrylic sheets don't put pressure on the cells. You can also use some sort of silicone adhesive to hold the cells down to one of the sheets.
As far as charging a battery, you probably want a boost converter to get the panel voltage to a battery charging voltage. Past that, it should just be a normal lead-acid charge circuit, there are several of these floating around the internet as schematics or off-the-shelf solutions.
Depending on your cells, you may not have enough power to really charge a big battery in any sort of reasonable timeframe, so that's something you may want to consider.