Hi!
Little bit of feedback in case someone has the same issues (although I'm sure there are less and less people transferring from dumb to smart phones ^^). Here is how I solved my problem, and it was actually not that complicated, I should have thought about it from the start… Only Bluetooth and SIM card required!
For the contacts, like I mentioned before, I used the "transfer via Bluetooth" option of the Contacts section on the original phone and it worked very well.
Messages took a bit longer, but I realized after transferring the SIM card to the new phone and going to Messages (which were empty) that if I tapped Menu and went to Settings, I then had "Manage SIM card", in which a bunch of messages were located. I copied them to the phone, but it was only part of the messages from the old phone, so I had to delete the ones on the SIM card, put the card back into the old phone and copy the messages to the SIM card one by one (the phone is so old that it didn't have any option to select all messages at once). Then move the card back to the new phone and copy to the device (this time, I was able to select all messages and copy, so the process was a bit faster). Delete on the SIM and then back and forth again between the 2 phones until everything was transferred (the SIM card's capacity is pretty limited). So yeah, a bit of a hassle, but if your messages are important, it's a solution!
The transfer messed up a bit some of the dates and times but I used the Android SMS Edit app to correct them
So there you have it!
PS: Since the original phone was not a smartphone, Samsung Smart Switch didn't recognize it, and I would have needed internet on the original phone to use a Cloud account…