How to copy and validate data from one table (all varchar) to another (typed) in C#?

Do it in T-SQL with a linked server.

I.e.:

--begin a transaction to wrap validation and load
BEGIN TRAN

--Validate that no tickets are set to closed without a completion date
SELECT * 
FROM bigTableOnLocalServer with (TABLOCKX) -- prevent new rows
WHERE ticketState = '1' /* ticket closed */ and CompletionDate = 'open' 

--if validation fails, quit the transaction to release the lock
COMMIT TRAN

--if no rows in result set 1, execute the load
INSERT INTO RemoteServerName.RemoteServerDBName.RemoteSchema.RemoteTable (field1Int, Field2Money, field3text)
SELECT CAST(Field1 as int), 
    CASE Field2Money WHEN 'none' then null else CAST(Field2Money as money) END,
    Field3Text
FROM bigTableOnLocalServer
WHERE recordID between 1 and 1000000

-- after complete, commit the transaction to release the lock
COMMIT TRAN

If you cannot communicate directly between the servers, still do the validation in SQL, but use a C# client to write the data to disk and hit the Bulk insert function on the destination server. Since the C# component would do nothing more than transport the data, I would just go straight to a format usable by BULK INSERT, à la CSV.