Do asynchronous event pushes count toward the 50,000 request limit in the GA API?
On the quota policy section of the documentation they describe contributing requests as "50,000 requests per project per day"
Is this only related to API calls or does it also apply to event pushes?
The 50,000 request limit is for the APIs both the Managemnt and Report REST APIs.
Data Collection is a different deal and have different limits. It is currently limited to 10M hits per month on Google Analytics Free Version. A hit is caused by any request to Google Servers as a result of tracking. and that includes pageviews and events among some less common like ecommerce, a social interaction tracking.
More info about data collection limits:
http://support.google.com/analytics/bin/answer.py?hl=en&answer=1070983