Tempdb is recreated with each restart.

There is no need to sync between nodes.

I have been doing this same thing with tempdb on my clusters for years.

If this is the approach you have to use, I think you should capture every login name, and every program name for every login. 
Examine the results of that after a...

Change the size of the batch in the SSIS package to a smaller number.

And, floow Jeff's suggestion to try to re-write thje query

You do not need to re-initialize.  Just add the indexes manually, and everything should be fine.

What RAID controller are you going to buy?  
The way in which you are describing things leads me to believe that you are not isolating drives onto different raid...

If you change the properties of the article, it is not going to add the indexes. You will need to reinitialize the subscription. 

Test it yourself.  Create a new...

is this the actual code?  If so, it looks like you are missing a "BEGIN".
Also, the %SQL % may be returning true for a number of different applications.  

In the properties of the article, you need to set the properties to include the indexes.  It does not do this by default. 
You can also create what ever indexes...

It's 70 and sunny. In Pittsburgh. In February. That's the reason.

You need to use single quote twice.
O''Brien

What YOU can do does not matter.  What matters are the permissions that the account being used to run SQL Server holds.

If the service account does not have...