can I do my statistics work based on the central limit theorem? I need to perform a t-test, ANOVA and multiple regression. my outcome variable is highly not normally distributed (Highly positively skewed) and my sample size N=115. I'd like to keep the non-parametric tests as a last option for me.
(1) The CLT is a result in the limit as $n\to\infty$. There's no particular
n that's certain to be large enough. e.g. see here which gives a method which works for constructing cases which require larger sample sizes than any $n$ you can nominate.
(2) the central limit theorem on its own is not enough. The statistics you mention rely on a ratio for which the CLT would only help with the numerator, and so you need something to hold for the denominator. The distributions also rely on independence of the numerator and denominator.
If the assumptions are reasonable, you might want to consider GLMs, perhaps (since with suitable software, regression via GLMs is almost as convenient as ordinary regression), though there are other alternatives.
There are various other nonparametric, parametric and robust alternatives.