Oracle 11.2.0.4: Basic info using RMAN for backups
QUESTION 1
So, how far back do I need to keep the archive log files?

You need to keep archive logs until you take another full backup.
If you take full backup weekly then you need to keep archive logs for week.

QUESTION 2
Does the database automatically remove the unneeded log files, or is this something I must address using cron jobs, etc.?

If you are using Fast Recovery Area for your backup and recovery purpose then Oracle will take care of it. Otherwise you need to delete it manually using RMAN commands.
For more:Configuring the Fast Recovery Area

QUESTION 3
To complete my understanding of archive logs, do I even need to put the database in archivelog mode to fully restore the database to the point in time of the RMAN backup, if I'm willing to accept loss of data from that time forward?

If you need to perform point time recovery then you need to keep archive logs, thus you are putting your database in archivelog mode. If not simply take cold backup and restore it you can not recover it since you don't have required archived logs.

QUESTION 4
When RMAN finishes, and I want to copy all required backup/restore files to another server, which files need to get copied?

You need to have control file and spfile backup which will be created automatically you have configured control file auto backup in RMAN and backup files to restore it into new host.

QUESTION 5
Not sure if anyone can help answer this, but I do a lot of scientific work that uses all available digits in a  binary_double number. If I choose compression for backups, will I lose accuracy when uncompressing the data?

I have never heard or read about inaccuracy in RMAN decompression. The only disadvantage that I came across is time taken to restore the compressed RMAN backup.