What is the grammar of "用餐曅請隨手整理桌面 座位有限請勿休憩估用"? In a cafeteria, there is a sign which presumably means something like, "There are a limited number of seats, so please don't sit down with just one person to a table; plan your seating arrangements so that everyone can have a seat."  I don't recognize most of the characters, but I managed to use a handwriting program to feed them into Google Translate, and I still don't really know what it is supposed to mean.

The actual text is as follows:

用餐畢請隨手整理桌面 

座位有限請勿休憩佔用

In particular, I can't figure out what the function of "畢" is.

There was no punctuation in the original.  I don't know if the division into two lines is supposed to indicate a division into two sentences or not.
As my paraphrase of the Google Translate shows, I can't even translate this phrase very precisely.

I suppose it can all be considered one sentence, but the "sentence" distinction doesn't really work as it does in English.   In English, the "sentence" is a very important distinction;  each complete thought is supposed to have a complete sentence. In Chinese, each paragraph seems to be one thought.

I know that 限請 is 'please don't' and '請隨手' means 'please make an effort, even though we can't specify in advance what measures you will need to take.'

用餐 might be "when you are eating" or it might be the adjective "dining" in "dining table."

I guess 座位有限 is "limited seating" or "a limited number of seats."

Can someone provide a grammatical explanation of how these vocabulary words go together?

Thanks.
This is probably easier with some punctuations:

用餐畢， 請隨手整理桌面。

座位有限， 請勿休憩佔用。

Let's break it down:

用餐畢: 用餐 means 'to dine' or literally 'to use meal'； 畢， short for 完畢, means 'to finish, to complete' (intransitive sense only); so altogether - 'after your meal'.

請隨手整理桌面: 請 - 'please'; 隨手 means 'while at it, while doing it' or literally 'to follow hand'; 整理 - 'to arrange, to put in order', in this case 'to tidy up'; 桌面 - 'tabletop' ; so altogether - 'please readily tidy up the table'

座位有限: 座位 - 'seat'; 有限, means 'having a limit, limited' ; altogether - limited seating

請勿休憩佔用: 請 - 'please'; 勿 - 'don't, not' ; 休憩 - 'to rest' ; 佔用 - 'to occupy'; altogether - please don't rest (here) and occupy (the seats)

The whole thing can be translated as:

Please tidy up the table after your meal; limited seating, so please don't rest here and/or occupy the seats.

Personally I'd have used 清理 instead of 整理, as 清理 implies cleaning and taking your mess with you whereas 整理 usually means to put things in order. It just sounds a little unidiomatic. Maybe it's more idiomatic in Taiwan (I assume this is where you saw the phrase from)?