Synchronous detector question
Yes, you can. You'd have to multiply your results with two.

But, you're multiplying with an on/off sequence in that case, and thus you'd obviously lose all info on the negative cycle of your input signal, and atop of that, reduce the amount of signal energy that reaches the integrator by a factor of two.

That reduces your SNR. And since you're building a lock-in amplifier, that's the worst you can do.

So, analog switches are cheap, and easy to acquire. Don't skip on them. In fact, the part that the author is using comes with four switches in one package, it's unclear why you'd want to use only one of them.