Can we have rests in the palette?

You can't drag notes from a palette to the stave. To enter a rest at the beginning of a measure press the duration (e.g. 5 for 1/4 note) and press 0. If you're using the mouse press the rest icon on the toolbar and you can enter rests that way. Press it again to continue entering notes.

"Rests exist where Notes are Not."

MuseScore knows that each Note is placed at a particular point in =time= (measure number + beat number), and defines Rests to occupy the silence. These periods of silence can be expressed using any combination of rests that will fit the duration of this silence. So, initially, "an empty measure" is occupied by (say) "a whole rest."

But you can =divide= that rest. Click on the whole-rest to select it, then click (say) the "quarter-note" button. The whole-rest will be subdivided into four quarter-rests. You can now place your new note where you wish.

In like manner, shorter rests can be converted into longer ones. Any combination of rest-symbols that fully accounts for the actual duration of silence in-between two notes (within a single measure) is fair game.

Another way of stating this:

Rests, just like notes, are entered left to right. Same way you read them. So to enter notes C D E followed by a rest, you type "C D E 0", but to enter a rest followed by notes C D E, you type "0 C D E". So as you can see, no special magic required to enter rests at the beginning of a measure - it works exactly the same as rests at the end of a measure, or as notes at the beginning of a measure.

If you prefer clicking instead of typing, you can do this also. Simply click the rest into the score before clicking in the notes. Again, it works exactly the same as entering rests after notes.

MuseScore4 Windows 11
Minor functional bug?
Click on a whole bar rest
Type 4 for quaver
Type 0 for quaver rest
Type C for quaver C to follow quaver rest. Result is bar starts with a C quaver, not a rest.

Work round
Click on a whole bar rest
Type 4 for quaver
Type 0 TWICE for rest
Type C. Result is bar starts with a quaver rest then C

I think needing to type 0 twice is a bug. If the bar is to start quaver C then quaver D you don't type C twice.