From an web app, we need to send around 1000 emails per month to different people. Is it a good idea to use smtp.gmail.com instead of Sendgrid, MailChimp or similar? Does Gmail impose any upper limit to the number of emails one can send?
If it is just 1000 total emails per month and you have a Google Apps for Business (not trial) account you should be fine. This page explains the various limits:
Sendgrid and Mailchimp have APIs that may make your code a little prettier, but at the end of the day if you are sending no more than 1000 a month total there is no functional difference in the delivery of email. Sendgrid and MailChimp offer more analytics about sent email and MailChimp has tools to manage email marketing campaigns, so if those features are important you may want to look into them. But if you just sending emails and do not care about that stuff you should be fine using Google Apps.