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One of the biggest mistakes on harmony theory was to assume that the rules that were originally conceived to vocal parts, to avoid problems related to a choir, people singing, had to be applied to every single thing can produce a sound. The SATB behavior was simply copied into strings section, keyboards and other instruments, and soon it became mandatory to any 4 part writing. I don't think many teachers will approve this I'm telling you, neither you will find it in books, but trying to care of every single interval as you were singing in a violin part, or piano or oboe or others is ridiculous. The doubling of the 3rd is to be avoided because this note, the 3rd, influences a lot the color of your chord, and doubling it will make a notorious overpowered 3rd, which is consider "wrong", other intervals like 5th doesn't seem to modify so much your overall balance, which you suppose to maintain equal in voices. In your example, you static chord in left hand already suggest you are not following the SATB rules, is only a piano part, since your rhythm is clearly not trying to follow the melody, (remember that if you were in choir, you also would have to care of lyrics). In a Piano part there's nothing wrong in that example, in a Choir part, the last two beats you are not only doubling 3rd but is also 8ved 3rd above your bassline, which indeed makes an ugly effect in choir, (maybe strings too). I'd say the only case where double+8ved 3rd can work (still against rules) is maybe in a 1st inversion of your chord as 8ved bassline, maybe. If you really want to write 4 voices and keep the rules, then keep reading, and continue to counterpoint, there is where you find how to move voices without breaking the rules.

"The modal degrees, by their insistence are hear by the ear as tonal degrees of another scale."-Piston. The explanation I've heard is that doubling the third can give it more importance to your ear than it is due, and lead to confusing the key to the listener. Since the key is established in our minds by hearing a lot of the tonic and fifth. If you hear too much third, your ear starts to assume the third is the tonic or the fifth instead of the third. Which can be an interesting effect to play with, but should be used intentionally and sparingly, or you just end up with mush. Nothing wrong with having a key, changing to a different one, changing back, peeking at a key and then not actually going there. That's all fine and lends interest to your piece. But there needs to be enough of an established key to create expectation in the listener's mind. Half of our pleasure in music is the anticipation of the continuation of a pattern. (And the surprise when the music doesn't go where you expect.) Got to establish it first.