What are the rules if you have two verbs in a sentence that are linked by "and"?
I've tried looking it up on the Internet but I'm not sure what the official term is.

For example:
They were eating and playing checkers.

I don't think this is correct. It sounds as if they were eating checkers and playing checkers.
Do both verbs have to be either transitive or intransitive, and do they need to share the direct object?

These sound correct:
They were eating and playing. (both intransitive verbs of the same tense)
They were eating food and playing checkers. (both transitive verbs with different objects)
They were cooking and eating food. (both transitive verbs with the same object)

Can someone please give me a link to a grammar site illustrating similar examples? Or at least give me the correct search words to find a grammar site?
I'm currently having an argument about this and need "proof".
This is an authoritative grammar site, Guest.

Compound verbs of this type must take the same grammatical form:

'To eat and to play checkers is my life.'
'Eating and playing checkers is my life.'
'I have eaten and played checkers, and now I am sleepy.'
'I will be eating and playing checkers when you arrive.'

If you find however, that it sounds a little strange-- 'I was eating and playing with the birthday cake'-- then you need to rearrange the elements:

'I was playing with the birthday cake while eating dinner.'
'I was playing checkers and eating.'

Personally, I find 'they were eating and playing checkers' perfectly clear. In writing, you most also presume that your reader has a normal sense of reality.
Joining like elements with "and" is called "coordination". "and" is a coordinating conjunction.

In your example, "They were eating and playing checkers", the semantics resolves the ambiguity of the syntax, i.e., "eating checkers" is rejected by reason of its strange meaning, so the ambiguous syntax does not create ambiguity of meaning in most normal conversations.

Note how the ambiguity is resolved in the reverse way in
"The committee asked and answered many questions."

I don't think it's a rule of grammar so much as it is a matter of how finicky one wants to be about resolving all such ambiguities explicitly - by rephrasing the sentence. Personally, I usually err on the side of resolving ambiguity wherever possible. At the same time one must assume at least a modicum of intelligence in one's reader! Emotion: smile
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Because they happened concurrently and independently for the time frame, you can put the verbs in any order. To remove any doubt about whether the object would apply to both verbs, put the verbs together with the objects ahead of any verbs which have no object. You can do it without adding or removing any words:
They were playing checkers and eating.
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