The first theme uses
D A F D or
^1 ^5 ^3 ^1.
The second uses
E E B B E B or
^1 ^1 ^5 ^5 ^1 ^5. (The opening phrase ends at bar 6. Bar 7 onward starts up a new phrase.)
The figures are similar - based on an octave and eighth notes, but the metrical placement is different.
The two are similar in that both outline a minor chord, at the same fast tempo. That certainly will give them a similar feel.
But in classical music this level of similarity is common.
Did the Beethoven 'inspire' the Dvorak? Seems reasonable, but I wouldn't call it plagarism. It's way, way more import to see how the theme is developed.
I dispute the idea that these passages are even similar, and from the perspective of 21st century copyright law in the US, this would probably not constitute "plagiarism" at all. In order to qualify as copyright infringement, these two passages would need to be so similar that a person without musical training would recognize them as the same.
These two passages aren't really that similar. It's true that the notes in the first four bars are the same (if transposed to the same key), but those notes are the tonic and dominant of the key - the most commonly used notes in Western music. The rest of the passage is wholly different, except that both end on the tonic (again, extremely common in Western music).
Lastly, the rhythms are completely different. Beethoven's rhythm is based off of a dotted quarter note, while Dvorak's is two eighth notes. Harmonically, Beethoven returns to the tonic in measure 6, while Dvorak delays it to measure 8.
It's certainly possible that Dvorak was inspired by this Beethoven passage, but it does not even come close to the standard of plagiarism.
Of course, US copyright law would have no influence over these pieces written in Europe, but this method of analysis is what's used to determine copyright infringement. If you're interested in learning about US copyright law, I recommend this website from the George Washington Law School: https://blogs.law.gwu.edu/mcir/ - It has information nearly all federal musical copyright cases in US history.
My memory is faulty to extreme, but...
A mid-20th century Eastern Euro or Russian composer (Prokofiev maybe) was disgusted enough with Shostakovitch's Bolero in his (Dmitri) 7th symphony that he (the disgusted composer) deliberately satirized the Bolero theme in a symphony of his own.
However, so far as NewWorld Symphony vs. Beethoven 9th, I am in the camp that says it's a weak equivalence in the first place and probably just a "likeable" rhythm for a Scherzo. I don't think the two pieces correlate.