[sql-server] Cause of a process being a deadlock victim

Q1:Could the time it takes for a transaction to execute make the associated process more likely to be flagged as a deadlock victim.

No. The SELECT is the victim because it had only read data, therefore the transaction has a lower cost associated with it so is chosen as the victim:

By default, the Database Engine chooses as the deadlock victim the session running the transaction that is least expensive to roll back. Alternatively, a user can specify the priority of sessions in a deadlock situation using the SET DEADLOCK_PRIORITY statement. DEADLOCK_PRIORITY can be set to LOW, NORMAL, or HIGH, or alternatively can be set to any integer value in the range (-10 to 10).

Q2. If I execute the select with a NOLOCK hint, will this remove the problem?

No. For several reasons:

Q3. 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?

Probably. The cause of the deadlock is almost very likely to be a poorly indexed database.10 minutes queries are acceptable in such narrow conditions, that I'm 100% certain in your case is not acceptable.

With 99% confidence I declare that your deadlock is cased by a large table scan conflicting with updates. Start by capturing the deadlock graph to analyze the cause. You will very likely have to optimize the schema of your database. Before you do any modification, read this topic Designing Indexes and the sub-articles.