If I am using my company's exchange server on iOS, can the company monitor other aspects of my phone usage such as browser activity, apps, or texting?

No.

It would be limited to the activities sent to the server (emails, calendar updates, tasks, etc.)

No, the exchange protocol isn’t that powerful or evil. It can only interact with your phone in the ways you let it, and only with mail/contacts/calendar.

If this is your personal iOS device you have nothing to worry about, if you’re using one that belongs to your company that’s probably a different story with Apple’s admin tools, but either way, having a linked exchange server isn’t going to be a backdoor.

As way of clarification. We’re talking about Microsoft Exchange here, correct?

If so, your company would only receive information that has to do with that Exchange account. So if you update your company email (send, receive, etc), they would get that information sent to the server, and the server would log it when it gets there just as it would an email you send from anywhere on their network. The same goes for your company calendar associated with your Exchange account and contacts, tasks, and other items that might be set up through that account.

Nothing else will be sent to the server, just like it wouldn’t for any other email account you have set up. If you don’t want to synch a certain aspect of the account with your phone/tablet, you can change the options in iOS under Settings -> Mail, Contact, Calendars -> <account> -> toggle on or off the items you don’t want to update.

It’s worth noting that if the phone is also company issued, or they pay your phone/data bill, they will get logs of your phone usage by number and time on the line, text messages by number, and data amounts used by session (so time and date for things like browser or app downloads). I’ve never requested more information. Note this is only if they pay the bill and isn’t anything more than anyone gets on their phone bill.

@sinscriven‘s excellent answer also just popped up at the same time, so I apologize for the overlap

If you are using the company wifi, they can monitor those things regardless of whether you connect to the Exchange Server or not.

Outside the premises, see the excellent answers above.