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Could anyone tell me all the issues or exceptions that we might have to handle in case of an integration?

Server down, network issue etc. 
Could you please specifiy the issues we would have during a callout?
All the issues please. 
Let's say: batch class and a callout, what issues would we have?
@future method and a callout: what issues would we face?
Callout from a flow: salesforce connect, external objects, record triggered flow (asynchronous path), outbound message?

VinayVinay (Salesforce Developers) 
It depends on your requirement on how complex the integration is,  however found below reference which mentions to avoid integration mistakes in future.

Please mark as Best Answer if above information was helpful.