Most likely.

I would not advise that. Covariance matrices are, by definition, positive (semi-) definite matrices and for the purposes you want to use PCA (which I'm assuming is some sort of substitute for Factor Analysis) you need that property. Otherwise your results are quite suspect.

One or more eigenvalues are negative and for true covariance matrices they are always positive (or 0, but that's another issue).

Not use PCA comes to mind. Perhaps doing more basic analyses using Classical Test Theory and then simply acknowledge you don't have the sample size to do anything else.

Yes. But nothing within SPSS will let you do that. You'd need to use a syntax-based statistical language (R, SAS, STATA, etc.) and learn some extra statistical methods so for your own and specific practical purposes I'd say no.