It's my first time DM'ing and we're running Lost Mines of Phandelver. 

The players have finished chapter 2 but this is where I'm winging it a bit as I'm not too sure on how it goes from here.

The players didn't

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>  save Sildar in Chapter 1. They went through the wolf cages and skipped it entirely so they don't even know he was in there. They don't know anything about Cragmaw Castle nor have they handed in their caravan quest.
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Is this going to cause any issues later on? 

Also, the 2nd chapter ends with

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>  them inside Glasstaff's quarters, and the 3rd chapter seems to begin almost immediately in the Spider's Lair. 
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Is there anything that needs to be done in between?