Simonas Invertus

The first questions that comes to mind is: do you use standard PrestaShop check-out or do you have one page checkout module? From PrestaShop perspective: it chooses default carrier in case it doesn't know what to show, so this is when free shipping comes out in most cases. If you have no free shipping carrier (even if it's disabled), there is no way PrestaShop could choose it. It comes to two possibilities: theme (as you said) or checkout/shipment modules. To understand what is happening, you need developer to investigate it. So my recommendation (maybe not very helpful) is to hire a freelancer or professional agency, there is no way to solve this issue without looking at the code.

You should contact module developers. It seems that images are imported incorrectly.

We you install new PrestaShop, PrestaShop generates random admin folder for security reasons. You can login to your FTP and rename it to anything you like as long as it's not "admin"

You can't create order in PrestaShop without customer details. I don't really understand your case, but it's not something that can be easily done, you will need to hire a developer to change check-out process which is hard work.

There are a few modules that do similar things you can find them here: https://addons.prestashop.com/en/search?search_query=tshirts&id_category=467& Contact developers and maybe they customize exactly to your needs.

It depends what you are using for shipping. If you using some kind of module, maybe that module has settings. PrestaShop allows you do have different pricing per zones, I am not sure if you can make a city a zone. It seems that this module could do the job https://addons.prestashop.com/en/shipping-costs/20858-table-rate-shipping-shipping-by-zip-code-kahanit.html I never used this module, so I am not sure about the quality, but you can write to developer and ask a few questions to make sure that it does what you need.

If I understand correctly, your server is using two step authentication process, which makes your email more secure, however PrestaShop or any automatic sign for that matter can't support two factor authentication. Solution: to find different SMTP server. Ask your hosting provider maybe they can provide one for you.

You should look into affiliate modules https://addons.prestashop.com/en/search?id_category=0&search_query=affiliate

There are a lot of options for marketplace, but only a few modules that do booking and I never seen a module that does both. I think the best option for you is: to pick marketplace module you like best and ask if developer would add additional features for booking for additional fee.

You can change order status via database, but you will need a script/cron job to send emails. Most likely you can use existing email templates and PrestaShop mail sending functionality, but you need to write a code that will execute it.

If you go to Traffic & SEO and scroll to the bottom you will find a line like this: Route to category {id}-{rewrite}. It means categories work only this id. In default PrestaShop ID is required, however some SEO modules can remove it.

Do you have the same issue only in FireFox or in all browsers? PrestaShop is using Ajax checkout, it means that if some of the modules crashes, you can't see an error message, you just see the same screen. I guess the best option would be to hire experienced developer to do the investigation. I don't think that there is something you can solve yourself. Unless you would disable ALL modules and see if that helps. If yes, then enable module one by one until you find the one not working. However I don't guaranty that even this way can help to find you a problem.

Is "Viewed Products" free module? If no, contact developer for support. It seems like an issue in theme, so there is hard tell anything without doing code investigation.

Not sure if I understand your problem, but did you tried to clear cache? You can find the button in Advanced parameters -> performance

Maybe the issue is that statuses are incorrectly translated? If you go to Shop parameters -> Order settings -> Statuses does it look normal to you?