What are the opinions that allow someone to shave and do laundry as per a regular weekday during Chol HaMoed? Is there a difference when it comes to shaving/doing laundry on Chol HaMoed for Yom Tov? What is the commonly followed practice?

The source for allowing one to shave on Chol Hamoe'd is Rabbenu Tam in the Tosafot to Moed Katan 13b as quoted by the Tur Orach Chaim 531. The logic of Rabbenu Tam is, that if the reason to not allow shaving is so that one should enter the Chag while unshaved, if he had shaved before the Chag then he should be allowed. The very same Tur who quotes Rabbenu Tam disagrees with him and says if that would be the case then the Talmud should have said so explicitly as it did in a case of one who has only one set of clothing who is allowed to wash his clothing during Chol Hamoe'd. Since the Talmud did not say so, he doesn’t see how it can be allowed. The Shulchan Aruch 531 rules that one is not allowed to shave on Chol Hamoe'd even if he had shaved before. If you are interested in more sources I suggest you see the Halachik journal Tchumin. In volume 2 there is an article which leans towards allowing shaving on Chol Hamo'ed if one shaved before. Though the conclusion of the author is that to begin with one should not shave, but you should fine merit for those who do. In volume 3 there is a response which explains all the reasons why it is wrong to shave and there is no way to disagree with the ruling of the Shulchan Aruch. To the best of my knowledge, the commonly held practice from what I see in Eretz Yisrael, is not to shave, and that is what I have been taught by my Rabbis