"I wash the tub and myself."
Translation:Labrum et me lavo.
In agreement with Idraote, one could try something possibly poetic with labrum meque lavo, in which case you have assonance of / l / and / m / with elision. Latin rhetoric and poetic meter allowed for repetition, so your use of lavo twice is not impossible and could be considered a deliberate amplification. See Quintilian, Insitutio Oratoria 9.3 "ad elevandum." On its own without poetic or rhetorical justification, it's an example of an unlikely Latinity.
You have to use se if it was third person for it to be reflexive. But for first and second person you would use one of the forms me, te, nos, or vos depending on who the subject is.
This link will likely provide a better overview than what I just wrote: http://dcc.dickinson.edu/grammar/latin/reflexive-pronouns
You may find helpful the following: http://dcc.dickinson.edu/grammar/latin/reflexive-pronouns