Yes, you can, but you'll struggle.
Jira has moved on a lot since that version and the plugins have either kept pace with it, or fallen by the wayside. If you're lucky, then a plugin that was around when 3.7.3 was a current version will have a historical version of the plugin that works for 3.7 in it's history and you'll be able to download that.
But a lot of plugins are more recent and not compatible back that far. I suspect you'll find you need to write your own if you insist on sticking with such an old version of Jira.
I'd highly recommend upgrading to a more recent version of Jira
Not a clue, as you've not told us what you're looking at.
The code for Jira 3.7.3 is still available if you have a current licence (in which case, why don't you upgrade to 5.2?), but for plugins, it depends entirely on if the writers have left it public at all.
And, of course, bear in mind that a lot of coding on plugins was started later than 3.7.3, so there simply was no code for that version.
That's technically another question, but...
1. Nothing, if you have a current valid licence
2. If your licence expired recently, talk to "sales at atlassian dot com" about getting it renewed
3. If your licence is too old, you will need to buy a new one. See http://www.atlassian.com/software/jira/pricing/?tab=download
You'll also need to budget time and costs of doing the work of course, but that's internal to your organisation.
Do you have any specific plugin on your mind. Please note that the most famous project management plugin which is Greenhopper only compatible from JIRA 3.13 and above. SO if you wish to use this plugin it is necessary for you to upgrade to at least 3.13.
Although, we always recommend you to upgrade to the latest JIRA version to ensure that you receive the latest feature and security advisory available for JIRA.
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