First off, I want to say that just because a company is big doesn't mean their security will be any better.
That said, I'll mention that having done security work in a large number of Fortune 500 companies, including lots of name-brands most people are familiar with, I'll say that currently 60-70% of them don't do as much as you'd think they should do. Some even give hundreds of third-party companies around the world full access to pull from their codebase, but not necessarily write to it.
A few use multiple private Github repositories for separate projects with two-factor authentication enabled and tight control over who they grant access too and have a process to quickly revoke access when anyone leaves.
A few others are very serious about protecting things, so they do everything in house and use what to many other companies would look like excessive levels of security control and employee monitoring. These companies use solutions like Data Loss Prevention (DLP) tools to watch for code exfiltration, internal VPN access to heavily hardened environments just for development with a ton of traditional security controls and monitoring, and, in some cases, full-packet capture of all traffic in the environment where the code is stored. But as of 2015 this situation is still very rare.
Something that may be of interest and which has always seemed unusual to me is that the financial industry, especially banks, have far worse security than one would think and that the pharmaceutical industry are much better than other industries, including many defense contractors. There are some industries that are absolutely horrible about security. I mention this because there are other dynamics at play: it's not just big companies versus small ones, a large part of it has to do with organizational culture.
To answer your question, I'm going to point out that it's the business as a whole making these decisions and not the security teams. If the security teams were in charge of everything, or even knew about all the projects going on, things probably wouldn't look anything like they do today.
That said, you should keep in mind that most large businesses are publicly traded and for a number of reasons tend to be much more concerned with short-term profits, meeting quarterly numbers, and competing for marketshare against their other large competitors than about security risks, even if the risks could effectively destroy their business. So keep that in mind when reading the following answers.
If source code were stolen:
Most wouldn't care and it would have almost no impact on their brand or sales. Keep in mind that the code itself is in many cases not what stores the value of a company's offering. If someone else got a copy of Windows 10 source, they couldn't suddenly create a company selling a Windows 10 clone OS and be able to support it. The code itself is only part of the solution sold.
Would the product be at greater risk because of this? Yes absolutely.
External Modification: Yes, but this is harder to do, and easier to catch. That said, since most companies are not seriously monitoring this it's a very real possibility that this has happened to many large companies, especially if back-door access to their software is of significant value to other nation-states. This probably happens a lot more often than people realize.
Internal Attacker: Depending on how smart the attacker was, this may never even be noticed or could be made to look like an inconspicuous programming mistake. Outside of background checks and behavior monitoring, there is not much that can prevent this, but hopefully some source-code analysis tools would catch this and force the team to correct it. This is a particularly tough attack to defend against and is the reason a few companies don't outsource work to other countries and do comprehensive background checks on their developers. Static source code analysis tools are getting better, but there will always be gap between what they can detect and what can be done.
In a nutshell, the holes will always come out before the fixes, so dealing with most security issues becomes something of a race against time. Security tools help give you time-tradeoffs but you'll never have "perfect" security and getting close to that can get very expensive in terms of time (slowing developers down or requiring a lot more man-hours somewhere else).
Again, just because a company is big doesn't mean they have good security. I've seen some small companies with much better security than their larger competitors, and I think this will increasingly be the case since smaller companies that want to take their security more seriously don't have to do massive organizational changes, where larger companies will be forced to stick with the way they've been doing things in the past due to the transition cost.
More importantly, I think it's easier for a new company (of any size, but especially smaller ones) to have security heavily integrated into it's core culture rather having to change their current/legacy cultures like older companies have to. There may even be opportunities now to take market share away from the a less secure product by creating a very secure version of it. Likewise, I think your question is important for a totally different reason: security is still in it's infancy, so we need better solutions in areas like code management where there is a lot of room for improvement.