I'm not aware of any options except the various roadside car parks and laybys.
Anyone know the current cost for camping at Gwern Gof Isaf or Gwern Gof Uchaf? As Kate says, they only allow day parking (Isaf certainly) but might it be an option to pay for camping and stick an empty tent up at the side of the car while you're off in the hills?
As it happens Mountain Hardwear has also worked for me. Back around 2005/2006 I found a pair of their Fusion Stretch Shorts which I guess today we'd call softshell - not cheap then at around £50 iirc, close fitting (but not totally figure-hugging like leggings or cycling shorts), hard wearing and with a fly, belt and 4 useful pockets. I was so impressed with them the first summer that I bought a second pair before they disappeared. The second pair are still in the drawer virtually unused, while the first pair have survived 8 years of walking and cycle touring and 5 years of pretty regular cycle commuting, and are still going strong.
Interesting that I'm not the only one to experience quality and longevity that made the price well worthwhile from that particular brand. Back in the 2000s I also walked in their Pack Pants which were equally indestructible. I don't know if they still live up to that as I haven't bought anything of theirs for several years now, and possibly the brand has changed, but a quick look shows they do still have 2 or 3 stretch shorts similar to the old Fusions in their range.
Yes, that strikes me as a fair description too. I've only done a few walks and cycles on Lleyn so far, but it's definitely got potential.
A lot of the cycling would be on quiet lanes. But at the northern end of the peninsula there's an excellent 12.5 mile cycleway - Lon Eifion Cycleway - which runs from Caernarfon inland to Bryncir, with good views of some of Snowdonia's hills. I used it on a tour last autumn - traffic-free, good surface, a few gates along the way but they're wide and "easy-open" so I don't think they'd be a problem with a trailer.
For walking the coast path is the obvious starting point, and tbh I reckon you stand more chance of finding the paths existing on the ground than you might with some of the other footpaths marked on the OS map.
There's a Cicerone guide - "Lleyn Peninsula Coastal Path - A Walking & Cycle Touring Guide" - which describes the coast path for both activities. It's divided into fairly short sections so could give some ideas for out-and-back routes or sections to look for a circular return (or possibly link by bus if walking).
Other than the walking and cycling, you'll have a choice of decent beaches, a few nice little villages such as Criccieth with its castle, and if you head as far as Porthmadog a choice of two narrow-gauge railways, the Ffestiniog Railway (to Blaenau Ffestiniog ), and the Welsh Highland Railway to Caernarfon via Beddgelert. Actually, either of those gives opportunities to link a train trip with a linear walk.
As for the supplementary question of "any other recommendations for an emergency winter kit for the car"...
I tend to put together a bag that stays in the car all winter. Among the contents (and this assumes I'll have a decent coat, hat and gloves with me anyway, and add food and drink if the journey warrants it) are: