The first note on the two-sided card is clearly Ashbrook, in his typical printed style and handwriting.
The second note is definitely "not" Ashbrook. The #652 note. That actually resembles Chase's handwriting and style. I suspect either @ioagoa or @classic can confirm or rebut that, as they will know I'm sure.
In the 5c book, Chapter 6 will further help you since your letter is going to France.
The '8' marking comes from France, and indicates 8 decimes due, for a letter weighing up to 7.5 grams or about 1/4 ounce. The 21c prepayment just got it to Britain - via an American Packet (Collins line). The addressee has to cough up 8 decimes for the transit from Britain to France.
From North Atlantic Mail Sailings page 105, your cover's route:
Departed New York Sep 5, 1855 on Steamer Pacific.
Arrived Liverpool Sep 15.
The orange marking on the back shows when it transited Britain (liverpool likely). The Black Calais marking on the front, shows when it entered the French mail system, and took the train from Calais to Paris (Sep 17). Then there is a Paris transit marking, and finally an arrival marking for Nantes on the reverse.
One way I like to think about British Treaty mail, is to consider that the British Empire at this time was large, and they had shipping routes all over the globe. So the concept, simplified, is if you are sending a letter to some foreign country other than Britain, then send it via British Open mail, so it will go to Britain and they will use their established routes to get it wherever it needs to go in the world. When it arrives at the destination, whatever additional postage is due beyond Britain will typically get assessed there - for example 8 decimes due to the addressee here.
The US did not have all of these connections so well established yet. The British Cunard shipping line was king of the Atlantic, so to speak, with US competition growing, but still much smaller. Because of this, mail carried on American packets, like yours, is sometimes a little more desirable, as its less common than mail carried by British packets. Except in 1855, when your letter was carried.

This is because of the Crimean war going on. The British diverted many of their ships to support the war, instead of going back and forth to the US (carrying mail). As a result, the American owned Collins line, carried a much higher percentage of the mail in 1855. You see a huge spike in the proliferation of 21c rate covers vs 5c rate covers in 1855 as a result.
Finally, your cover is part of the Garnier correspondence. This is a very large and well-known correspondence. Your cover shows no evidence of fakery, but for your amusement and further education, you can see a write-up of one of the more famous fake covers of recent times - the Klep cover here. It is a Garnier cover.
https://www.rfrajola.com/opinions/klep.htmedit: Oh yes, and Ed Loquet was a very prolific forwarder.