A large number of people gathered outside the embassy - how many
We use a large number of when we want to indicate how many.
Many people gathered outside the embassy - how many
Many people believe that the commng year will be difficult - what proportion
We can use many to indicate either how many, or what proportion. As a proportion, many can used for about 40 to 80%, so it fits between quite a few and most.
When we want to indicate how many, we more often use a large number of because it is clearer than many, which has two possible meanings.
If you use a large number of, you have to use a plural noun- men. You would only use a singular noun to describe a quantity when there is only one, or the noun is uncountable, for example sugar. For non-count nouns, you would use amount rather than number.
A large amount of sugar.
You can't say "lot of": you can say "lots of" or "a lot of" . Both have exactly the same meaning as "a large number of" but are less formal. Even less formal are "loads of" or "a shed load of". You can use these informal expressions with countable nouns eg people or uncountables eg sugar.