How should one play this passage in Beethoven's Op.2 No.1? One cannot play staccato and use the pedal because notes on the upper stave would not sound staccato.
Slurs are not pedal marks. In this case I would interpret the slurs as phrasing slurs, meaning that while playing all of them staccato, one bar forms a phrase of them. A phrase can be executed by a consistent execution, like a subtle development in volume or staccato shortness or accent that repeats every measure.
When the bass line is taken by a bowed string instrument (obviously not the case for a piano sonata), this kind of notation means to change bow direction every bar even while every note is staccato in itself. The musical effect of tying the notes of one measure into a phrase then has a more tangible expression. Maybe entertain a mental image of that when playing it on the piano.
The bow or slur above the staccato notes is one type of portato notation, also used for staccato and flying spiccato.
Not only for strings, for all iinstruments and also for pianos this slur means the same articulation, but it is not a legato and is not played with a pedal:
The desire of editors and composers to make their intentions clear down to the very last detail includes the use of large sweeping 'slur-like' lines called 'phrase marks'. Slurs, which tend to embrace a smaller number of notes, help to shape the musical line even within broader phrasing marks and performers must be able to distinguish between them. On wind instruments, all the notes under the slur except for the first, are untongued, the breath flowing continuously while the fingers move. On stringed instruments, the equivalent effect is achieved by using a single sweep of the bow for each slur or phrase. On keyboard instruments the notes are played legato (smoothly) and with a light touch. The slur removes the attack from the start of each note under it except for the first so providing a contrast in strength, a dynamic variety, between the first and the later notes. If slurring is to be effective, or indeed a distinction made between different phrases, the performer must avoid playing unslurred notes too smoothly. Evidence from the eighteenth century suggests that music then was played in a more detached manner than we associate, say, with the repertoire of the late-Romantic.
The way a staccato mark under a slur is realised will also depend on the instrument for which the instruction refers. On a piano, the staccato under a slur is a portato where the individual notes sound for three-quarters of their written duration.