What happens when a node goes offline for a while? Let's imaging a full node goes offline for a month, so they're missing ~4300 blocks (just using 10min/blk). 


Are there any message exchanges that receive transactions to help sync the 'booting' node with the other peer node's mempools? (I.e. compensate for having a empty tx mempool)
Approx. how long it take (presume the 4300 missing block ex.) to receive the missing blocks?
What is happening during this time if the node is receiving propagated transactions? Or even a newly solved block?


Thanks!
Full nodes are always in a perpetual state of "catching up to the network".  This is true if they are just booting up for the first time, have been down for a month, or have been running since the network began.  Nothing is fundamentally different about their operation.

The amount of time a node will take to catch up is determined mostly by a) how many blocks/transactions it still needs to verify, and b) how fast the node is at verifying blocks/transactions.  To a lesser extent, it is also determined by network speed (how fast it can receive blocks/transactions), but the bottleneck is usually the computational power of the node.