Actually, you can. it's called כתיב חסר/Ktiv Haser (spelling without vowls). It is somewhat confusing, and now days, since we almost never use ניקוד/Nikud (Visual Punctuation) in modern hebrew, it is almost never written in this form. בקר means 'Cattle' and ביקר which can be written בקר is 'visited (m.)'. Hope I helped!

No, we can't because it lead to misunderstanding and can easily confuse with בַּקָר= visit

For practical everyday use, if you mean "morning" always put the vav. Otherwise it looks like "cattle".