Need clarification, brass instruments, pre-1900 score

Please do everyone a favor and transpose these parts. No need to carry on an outmoded tradition.

Indeed there are horn players that read "no key" parts. But it is not universal. If you modernize a score, more people will be able to play and enjoy it.

At the time in question, horns did not have valves.To play the harmonic series in a particular key, (let's say "E") the player installed their "E" crook. The player had 8 or nine different length crooks that they would change in order to play in different keys.
Same with trumpets. Which is what the "trombe" is.

Don't mess with the part properties. Just select the appropriate instruments from those available in the dropdown list when you set up the score. In your case you would have two horns, one in E and one in C.

"Trombe" is actually trumpet. So select a trumpet in E from the list. There is a similar tradition of no key signature for early trumpet parts, but I think that tradition died out earlier than the horn tradition. If there is no key signature on the Trombe in E part and the rest of the ensemble has anything other than four sharps, then the score you are looking at is following that "no key signature" tradition and you should again use the "open" key signature.

Sorry, but this quote is probably one of the most ridiculous, dangerous, incorrect things I have ever heard. Europe (she worked for a British company) has along tradition of no key signature. This is sadly true.
Two hundred years ago, composers wrote for the instruments they had. There were no trombones or tubas. There were sackbuts and serpents. Trumpets and horns had no valves. Modern orchestras don't don't have players using any of those instruments. True that modern trumpet players will tend to use C trumpets for most things. And to reproduce Bach parts, players may use a high pitched instrument. And sure, there are players using these "natural" (valveless) instruments. But those are few and far between. And even concerts of players using reproduction instruments aren't very authentic because we just don't know how these instruments were played. Instruments aren't the same today. They don't look the same, nor do they sound the same as they did originally. There is only one original Stradivarius violin in the world. It gets played once a year. All the rest have been heavily rebuilt, re braced, and had longer necks installed so that they can play in modern orchestras. It is possible that Beethoven might not recognize his own music if he heard it today. Even so, transcribing and old piece and not making it accessible to all modern players just seems a bit "academic", less approachable.
And having played trumpet in many orchestras, I never saw a no key signature part. OK once. We were playing Handel's Creation and the part was in "D". So for one rehearsal I played my Baroque trumpet in D. No one Noticed or batted an eye. Even the conductor.