Cause of a process being a deadlock victim

I have a process with a Select which takes a long time to finish, on the order of 5 to 10 minutes.
I am currently not using NOLOCK as a hint to the MS SQL database engine.
At the same time we have another process doing updates and inserts into the same database and same tables.
The first process has started, recently to end prematurely with a message

SQLEXCEPTION: Transaction was deadlocked on lock resources with another process and has been chosen as the deadlock victim.

This first process is running at other sites in identical conditions but with smaller databases and thus the select statement in question takes a much shorter period of time (on the order of 30 seconds or so). In these other sites, I don't get the deadlock message in these other sites. I also did not get this message at the site that is having the problem initially, but, I assume, as the database has grown, I believe I must have crossed some threshold. Here are my questions:

1. Could the time it takes for a transaction to execute make the associated process more likely to be flagged as a deadlock victim.
2. If I execute the select with a NOLOCK hint, will this remove the problem?
3. I suspect that a datetime field that is checked as part of the WHERE clause in the select statement is causing the slow lookup time. Can I create an index based on this field? Is it advisable?