Is the construction 连A带B meaning "both A and B" a general construction or limited to fixed expressions?
Can it be used to link anything together, or is it only used in fixed
expressions?

Using [連 verb A 带 verb B] create mostly idiomatic phrases when the verbs are related e.g. 消打, 哄骗, 滚爬
连消带打, 连哄带骗 and 连滚带爬 are considered fixed expressions because they are commonly spoken in day-to-day speech.
But A and B in [连 noun A 带 noun B] can be anything related or not

In fixed expressions A and B are both verbs. Does it only work for verbs?

It can be nouns, e.g. 连人带马一起翻落山坡; 连店带货一起顶让给别人

I've found what appear to be examples of 连A带B带C to connect 3 things together. Does it work for as many things as you want?

The common structure is (连)A(带)B(与及/外加/再加上)C. e.g. 连人带马与及军旗一起翻落山坡; 连店带货再加上招牌一起顶让给别人; 連哄帶騙外加小許威迫

Does it add any emphasis or other flavor to A+B? Simultaneity? Intensity?

连A带B indicates 'also' or 'both'.

verb A and verb B happen at the same time or alternately.

noun A and noun B are both the objects


Emphasis is on 'about equal in importance'