The negligible eccentricity of all extrasolar planets with periods less than 6 days can be accounted for by dissipation of tidal disturbances within their envelopes that are induced by their host stars. In the period range of 7-21 days, planets with circular orbits coexist with planets with eccentric orbits. These are referred to as the borderline planets. We propose that this discrepancy can be attributed to the variation in spin-down rates of young stars. In particular, prior to spin-down, dissipation of a planet's tidal disturbance within the envelope of a sufficiently rapidly spinning star can excite eccentricity growth and, for a more slowly spinning star, at least reduce the eccentricity-damping rate. In contrast, tidal dissipation within the envelope of a slowly spinning low-mass mature star can enhance the eccentricity-damping process. On the basis of these results, we suggest that short-period planets around relatively young stars may have a much larger dispersion in eccentricity than those around mature stars. We also suggest that because the rate of angular momentum loss from G and K dwarfs via stellar winds is much faster than the tidal transfer of angular momentum between themselves and their very short (3-4 days) period planets, they cannot establish a dynamical configuration in which the stellar and planetary spins are approximately parallel and synchronous with the orbital frequency. In principle, however, such configurations may be established for planets (around G and K dwarfs) with orbital periods of up to several weeks. In contrast to G and K dwarfs, the angular momentum loss due to stellar winds is much weaker in F dwarfs. It is therefore possible for synchronized short-period planets to exist around such stars. The planet around Tau Boo is one such example.