How do I put a gliss on two notes, one of which falls on a downbeat, the other on the third beat of the next measure, without having a rest automatically put into the space without notes (beats 1 and 2) in the second measure?

And how do I go about tying (?) a half note to another half note, but putting a gliss on the second half note without Dorico deleting the gliss?

Avoiding rests in the 2nd bar: you probably want to make sure that whoever’s reading it knows how many beats the gliss passes over, so maybe consider using headless notes and showing two crotchet stems on those beats. (There are other methods available for that, they should come up either in that thread or elsewhere on the forum). Dorico will show implicit rests automatically unless you remove them, but in this case I wouldn’t recommend that as in a part layout (with only one staff), without the rests to use for spacing, the first half of the bar might end up very narrow and the half note on beat 3 might look like it’s at the start of the bar.

Tying notes: if you’re in 4/4 and want to have two tied half notes, make sure both have Force Duration active (select them and press O) before tying them left-to-right. It is likely that the gliss will get deleted anyway, as Dorico considers tied notes as a single “note unit”, which technically deletes the second note that had the gliss, replacing it with an elongated first note which didn’t. You can just re-input the gliss though - it will appear attached to the last note in the tie chain.