I’ll preface this with while I do run a successful web store and have been doing so for some time I’m in no way an expert and am still pretty crap at technical tweaks. Also I have a habit of doing walls of text so this will probably be long.
To start most of your website issues are probably related to the store system itself, presta shop in your case, which for things like this you really have to be knowledgeable about that system in particular to get things like checkout and such setup. You probably already have been looking on presta’s forum and if it’s anything like Zencart you probably can find someone who has done exactly what you want and maybe even made an addon. Besides a few visual tweaks like the logo background which I think you just need to bring that image into gimp and change the white to an alpha channel I would aim for practical things like shipping estimator, more shipping options, payment options, clear information about shipping and payment options (more details below). The site being easy to use and usable by buyers with a wide range of needs is IMHO more important than looks so best to do those first.
For payment you may want to keep just PayPal, for a long time I just used paypal and really didn’t have many issues but you have to make it super clear that paypal accepts credit cards. Paypal doesn’t exactly help with this since they tend to push paypal a bit on the payment page but trust me if this is not super clear through the whole checkout process people will give up or email you asking about credit cards. In terms of other payment methods like a direct credit card payment processor there are two categories here it seems to me, easy setup ones like stripe and all those ones they advertise on TV and full on ones like authorize.net. The main difference is easy ones are easy to setup but you pay more, big ones can be tricky to setup have super complex fee structures and can be cheaper. Either way I would stick with offsite payment handling. Like Paypal the customer leaves your site, pays, then returns. This means no finanical data is ever on your site which takes a ton of security issues away from you, makes PCI compliance much easier (which you will need for big boy processes and maybe for easy ones). I did a ton of searching and big processors won’t tell you their fees upfront but after you do get through them, they are a little cheaper than paypal if you have the volume. Really paypal is probably the best way to go at least for awhile, they are easy, secure, and pretty cheap.
Shipping it a tricky one and still one I struggle with, you want to be clear on shipping times and offer a wide range. USPS is great but there time estimates are often totally off, things pretty rarely get lost but often delayed a week with no useful tracking. Offering something like FedEx gives a more reliable (although more expensive) option to those that want it and may be the way to go for more expensive items. I don’t even want to get into the mess that is international shipping. As stated above a shipping estimator is a very nice features just as a user.
At this point I’m convinced that product photography is an art and a science to rival any other. I’ve gone through so many iterations of my setup I’ve lost track. Now I run, 6x 1600 lumen 5000K CFLs, 4 from above with diffusers, and 4 from below with a diffuser (glass stage so things are lit from below, no shadows). Coupled with that I now have a Cannon 450D on a tripod with remote shooting. Frankly for a long time I used a ancient Olympus C-4000 zoom and the pictures were pretty comparable to the Canon if you get the settings right. I’m no photography expert either but if you have a point and shoot you need a tripod and you need to use it with the self timer. In addition I set the F stop as high as it will go (again with a point and shoot it won’t go very high) and then take 2-3 shots changing only the shutter time (around 1/10 to 1/25). Then I pick the one I like the lighting best and maybe tweak just a hair in gimp. The color of the object makes a big difference flat black items are pretty easy while super white or clear objects can be a pain. Small objects use the macro mode. The thing that I haven’t seen many people use is the bottom lit stage. For me this is a cardboard box, lined with tin foil, with two lamps in it then on top goes a piece of glass with a sheet of teflon sheet on top. You don’t need to use teflon of course I just happen to have it. Any decent diffuser material will work. You may need to balance the lights above vs below (I use 2 below 4 up) but after you do that the shadows are gone. I can show you an example if you like.
And the TL:DR version is stick with paypal, lots of checkout directions, more shipping options with details about them, cheap camera is fine but tons of light from all directions, good settings and a tripod. Any questions feel free to ask.