During the initial data run with the High Resolution Spectrometer (HRS) at SLAC PEP, an integrated luminosity of 19.6 p${\mathrm{b}}^{-1\mathrm{}}$ at a center-of-mass energy of 29 GeV was accumulated. The data on Bhabha scattering and muon pair production are compared with the predictions of QED and the standard model of electroweak interactions. The measured forward-backward charge asymmetry in the angular distribution of muon pairs is -8.4%±4.3%. A comparison between the data and theoretical predictions places limits on alternative descriptions of leptons and their interactions. The existence of heavy electronlike or photonlike objects that alter the structure of the QED vertices or modify the propagator are studied in terms of the QED cutoff parameters. The Bhabha-scattering results give a lower limit on a massive photon and upper limits on the effective size of the electron of ${\Lambda }_{+}>121\mathrm{}$ GeV and ${\Lambda }_{-}>118\mathrm{}$ GeV at the 95% confidence level. Muon pair production yields ${\Lambda }_{+}>172\mathrm{}$ GeV and ${\Lambda }_{-}>172\mathrm{}$ GeV. If electrons have substructure, the magnitude and character of the couplings of the leptonic constituents affects the Bhabha-scattering angular distributions to such an extent that limits on the order of a TeV can be extracted on the effective interaction length of the components. For models in which the constituents interact with vector couplings of strength $\frac{g^2}{4\pi }\sim 1$, the energy scale ${\Lambda }_{\mathrm{VV}}$ for the contact interaction is measured to be greater than 1419.0 GeV at the 95% confidence level. We set limits on the production of supersymmetric scalar electrons through $s$-channel single-photon annihilation and $t$-channel inelastic scattering. Using events with two noncollinear electrons and no other charged or observed neutral particles in the final state, we see one event which is consistent with a simple supersymmetric model but which is also consistent with QED. This allows us to exclude the scalar electron to 95% confidence level in the mass range 1.8 to 14.2 GeV/${\mathit{c}}^2$.

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