Is 一月, 二月, 三月, et cetera the actual words for Chinese months of the year?
When I type in these characters I get the equivalent month for the English calendar. Meaning for 四月 I will get the translation for April, 五月 for May and so on. Do these characters only represent English months of the year or is it standard for Chinese people to speak about months using numbers?

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yes
sometimes you can see them as 月份
and
周一，周二， etc are real days of the week, Monday, Tuesday etc

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We use the numbers from 1 to 12+月（yuè）.
BTW: In China, we also have the lunar calendar农历（nóng lì）. We usually use 正月and 腊月.
正月（zhēng yuè）--the first month of the lunar year
腊月（là yuè）---the twelfth lunar month

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It is how we Chinese name our months, number + 月。 Sometime we also say number + 月份。You see, Chinese is easy meh! :)

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The same for the days of week. 星期一 or 周一means monday, literaly "first day of week". No special names, just first day of week, second day of week, etc. Only Sunday is called 星期日which means Sun of the week or Day of the week.

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yes. the word yue4  月  mean  moon month

so  chinese present month as

moon 1
moon2
moon3........moon12

just like if you says
first minth
second mth
third mth.......twelth mth

but on top of this  also use  
january............disembre