The first question has been discussed extensively in the literature. Some propose using the AIC or SIC criteria, others employ significance tests. To cite just a couple of references, Ng and Perron (1995) compared methods based on information criteria with sequential testing of additional lags. Ng and Perron (2001) propose a modified information criterion.
I agree with @ChristophHanck that there isn't a single strategy that will perform better than others in any possible situation. The residuals of the Dickey-Fuller regression should be inspected instead of trusting the choice of a given procedure. If serial correlation remains in the residuals, then one additional lag can be added until no structure is detected in the residuals.
Bootstrap is an interesting alternative because it can potentially deal with any kind of autocorrelation beyond those structures explored by simulations in the literature.
Note: P-values based on the original critical values may be a rough value. The distribution of the statistic has been tabulated for some fixed sample sizes and do not account for serial correlation. Cheung and Lai (1995) employed the method described in MacKinnon (1996) based on response surface regressions to obtain a procedure that gives the p-value of the ADF test for different sample sizes and lag order selection methods.
Cheung, Y. and Lai, K.S. (1995)
Lag Order and Critical Values of the Augmented Dickey-Fuller Test.
Journal of Business & Economic Statistics, Vol. 13, No. 3 (Jul., 1995), pp. 277-280. URL
MacKinnon, J.G. (1996)
Numerical Distribution Functions for Unit Root and Cointegration Tests.
Journal of Applied Econometrics, Vol. 11, No. 6 (Nov. - Dec., 1996), pp. 601-618. URL
Ng, S. and Perron, P. (1995)
Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag.
Journal of the American Statistical Association, 90(429), 268-281.
Ng, S. and Perron, P. (2001).
Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power.
Econometrica, Volume 69, Issue 6, pp. 1519-1554.