The first edition (published 1852) had staccatissimo dashes on the first two bars, not staccato dots. You can find it on IMSLP.
Old music editions usually assume that the performer had some musical common sense, and in this case (as you guessed) that means the articulations apply until either the music changes or there is an instruction to stop playing them....
This is where using more than one voice comes into play.Split the bass clef into two voices one with stems pointing downwards, the others pointing upwards.
It's a common thing to do, when effectively there are two separate things happening simultaneously. So the A note underneath will be for three beats, and its stem facing down, and there's a rest above ...
OI vopiSeveral posts down, there is an example of correct notation.
I copied it from that post. One notates the sound as two voices (thus the quarter rest) with a dotted half in one voice and a quarter rest and two quarter notes in the other voice. It's a common notation when the keyboard plays more than one voice in each hand.
Are the first four staccato marked based chords an implication to play the rest as such?
Yes. Your intuition is correct. The accompaniment's texture shouldn't change drastically when the melody enters. Burgmuller isn't that subtle. The marked scherzando also implies bouncy rather than lugubrious.
Is this implied staccato common practice in sheet ...
Opposite to other answers I would suggest that when the right hand entries (leggiero) the accompaniment should be played less staccato or less marcato.
The composer could have marked simile if he had wanted to continue the style. I think it is quite natural or musically logical to play the intro different than the accompaniment of this light melody.
If you really want this, then put the dot just under the actual played note, not under the tied-over one.
[V:T1] ([DB]2 [Gc]) z .[Fd]2 z .[DB]-| [DB] z .[B,DA]2 z .[B,FG]2 B,
Having multiple dots within a note, even a tied one, would imply you actually want multiple attacks on it. This is ...