Okay, so let me answer my own question. Remember the constraints: EU citizen customer, company shipping to a drop point, lens has insurance or damage waiver.

First of all, I have reviewed several online companies, but only LensRentals (lensrentals.com) accepted me as a customer from the EU. (Quick side note about BorrowLenses.com: they do not accept orders placed with foreign credit cards without a US billing address, so if you have one with a US billing address, you should be fine.)

Back to LensRentals.

I ordered a Tokina 11-16mm f/2.8 AT-X Pro DX for Nikon DX. They required some data to verify my identity, these were:

1. Work or school email address;
2. Employment information including the name of my employer, job title, and a work number I was reachable at;
3. Personal or business website;
4. A scan of my passport (must be included).

I provided these, and they accepted my order. (The scan of passport does not show up in the initial questionnaire, so I got an email from them asking for that).

In fact, I could not provide a business website, so I provided a personal website, my photo.stackexchange.com user profile page! ;-)).

They offer a damage waiver, which is not a full insurance, but it limits your costs to 10% of replacement or repair (NOTE: read their fine print so that you understand what is being covered and what is not. It is not too defensive, but you do not get full insurance! I am very nice with lenses, I do not even use front element protection, yet all my lenses are flawless, and people trust me with even the most expensive lenses of theirs).

They charged the full amount on the day the lens was shipped. They offer extension if possible, and you can also send back earlier if needed - there is a fee for that though.

You have 24 hours after reception to report if your lens is damaged, and they will try to replace it ASAP.

My lens was in a very good condition, no scratches, completely clean and working fine. The protection is about 1.5 cm foam poured into the box, and the lens is also in the original cover AND bubble wrap. You have to keep these and use them when you are shipping back. You are provided with a shipment label, so I just glued that over the old one, removed the other labels (important!), packed the lens carefully and took it to a FedEx point.

Now, I was careful to choose a FedEx drop point (checked many on Yelp) that seems to be very reliable, because they do not ship to a hotel address. I used the same FedEx shop to ship back the lens to them.

They ask for a full day after the lens arriving back, so that they could inspect, clean, and see if something is missing, or something is extra (e.g. a memory card).

I am perfectly happy with this service, I made great photos, and the only thing that is missing is a confirmation at the end that the lens has been received back in good condition and the transaction is complete. Otherwise you can track your lenses online, change order before and during the rental, etc. Chat/support is great, too.

I will put up some pictures of the packing quality and the lens condition, if anyone is interested.