We already had tanto in all its variants, see 28.1. The meaning can be so many, so much, so many things. The difference between so many / so much / so many things and a lot of is the same in Portuguese as in English, although a lot of doesn't distinguish between quantity (so much) and number of elements (so many).
a lot of <=> so many
1 a) I didn't expect that so many people would come.
I didn't expect that a lot of people would come.
a lot of <=> so much (adjective)
1 b) One doesn't need so much money.
One doesn't need a lot of money.
a lot of <=> so many things
1 c) There are so many things one doesn't know.
There are such a lot of things one doesn't know.
a lot of <=> so much (adverb)
1 d) Don't smoke so much.
~ Don't smoke a lot.
From a purely logical point of view there is no difference between so many / so much / so many things and a lot of. Implicetly we have a comparison in the constructions so much / so many, but the thing we compare with is not mentioned. (In 1a, b, c, de we only know that the amount exceeds a quantity, but we don't know which quantity he has in mind and therefore so much, so many is as relative as a lot of. That doesn't make, from a logical point of view, any difference.)
Concerning muito there is a choice when it describes a big amount. It can be translated with a lot of or many. If it describes a big quantity of something the only possible translation is a lot of. Muitos can be used as a noun as well, in this case it has to agree in gender and number with the noun it stand vor. If there is nothing it refers to, "There are a lot of things I don't know" <=> "Há muitas coisas que não sei". (We have to add coisas.)
Muito con be adverb as well. In this case it is translated with very.
muitos / muitas pronoun
Tem muitos que se crêem muito espertos.
There are a lot of people who believe to be experts.
Tinhas uma coisa muito especial que muitas não tinham.
You have something very special that not many people have.
muitos / muitas adjective
Parece que muitas pessoas trabalham neste mercado.
It seems that a lot of people are working in this market.
Muitas raparigas trabalham em mais do que um sítio.
Many girls have more than one work.
Já vi muitos homens como você.
I have already seen a lot of people like her.
muito / muita noun
Parece que tem muito no seu prato.
It seems you have a lot on your plate.
Não tenho muita.
I don't have a lot.
muito / muita adjectivie
Podíamos ganhar muito dinheiro com isso.
We can earn a lot of money with that.
Deve haver muita água lá em baixo.
There must be a lot of water beneath.
Trabalhamos muito para construir esta unidade.
We work hard to form a team.
É muito estranho.
That's very strange.
If muito refers to a group of heterogenous elements, "There are a lot of things he doesn't like", the concordance in gender and number is difficult, because the different elements have different genders or the elements are simply unknown. In this case we have to use muitas coisas. Coisas can be everything, that always fits.