How do I turn the highlighted E into a quarter note? I succeeded doing so for the D after it, but the E stubbornly insists on staying eighths notes.
Similarly, how do I turn the following ugliness into a whole note?
For the first one, use the Force Duration tool. The icon looks like a clamp.
More info here.
For the second one, alter the time signature to 8/4, I guess…?
Go into the notation options and check the note groupings sections. You’ll find global settings for these.
How do you want the second one to be a whole note if it is crossing a barline?
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What I normally do in these cases: I select the note, then hit consecutively „O“ „5“ „6“
Then I would select the other note and do exactly the same.
You mean O, right?
Oh, yes 
o-right leo, I am at my phone just now…
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For the second one, you either need to increase the note’s duration by an eighth note - so set the grid duration to an eighth (Alt [ or ] ) and then Shift Alt Right Arrow to lengthen the note.
(Or move the entire note one eighth forward with Alt Right Arrow.)
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For using Force Duration, the most recent version of that topic in the 3.5 manual is here (unfortunately it’s still quite easy to land on old manuals from online searches).
However, in general you’ll probably find it quicker and easier in most cases to use (if you have Dorico Pro) the available note grouping options, potentially in conjunction with specifying the beat division as part of the time signature.
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Thanks everyone for helping me out! Here’s how I solved the problems:
To turn the two eighths notes into one quarter note: Select the note as shown. Change it into an eighth note (which is then followed by an eighths pause). (That’s the step I’ve been missing.) Click the Force Duration icon (or press O), then change the eighth note to a quarter note.
To change the eighth note with the following mess in the next measure to an eighth note plus a whole note (sorry for not being precise here): Select the messy notes, and the eighth pause at the end. Select Write > Edit Duration > Extend to End of Selection. That will do the trick:
To be honest, I find that a bit cumbersome. In MuseScore, I’d simply select the beginning of the “notes mess” in the new measure, then click Whole Note. Having to adjust stuff like that is a common task when writing syncopated music, so I think this should be easier.
Anyway, thanks again for your assistance!
The difference is that Dorico thinks of that long note as a single duration, so telling it “this is a whole note” isn’t actually true: it’s a note that is nine eighths long.
One of the advantages is that if you change your mind about the note’s position in the bar, or the time signature, Dorico will automatically regroup the note. One of the other advantages is that if you apply an articulation to a passage that contains a tied note, the articulation will only appear at the correct end of the tied note. Yet another advantage is that if you want to build a chord, Dorico will automatically make all of the notes in the chord the tied length.
I admit I’m running a slightly older version of MuseScore, so I may be out of date on its capabilities here, but, for instance, if I put a half note on the fourth eighth of a 4/4 bar, Dorico gives me an eighth tied to a dotted quarter (which is correct). In MuseScore I think I’d need to enter an eighth and then tie it to a dotted quarter, manually, which is considerably more effort.
In your case, you could have set the rhythmic grid (bottom left corner) to eighths, then selected the long note, then typed Shift-Alt-Right.
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There is more than one way to increase the length of a note. I’ve already described one way; you have found another.
Thanks Piano Leo! That’s an easy way of making notes longer (or shorter). Excellent! (Forget what I said about “cumbersome” 
)