Yes! And this is what's critical to the success of the one-time pad: you've not just encrypted data, you've encrypted data in such a fashion that (from the attacker's perspective), all plaintexts they recover are equally likely. Cryptanalysis is fundamentally impossible.
The information theory is absolutely groovy, but the application of it is unfortunately impractical.
Not that it matters. The sqrt(n) birthday effect isn't sufficient to overcome the exponentially low chance of a one-time pad collision for pads of any substantial size.
And to expand on this - this is why it is imperative that a source of true randomness is used as a seed. Pseudorandomness cannot (by definition) satisfy statistical independence.
If you don't have a source of true randomness, you've just implemented a stream cipher, not a one-time pad.
You're not likely to get a collision ever because of the typical sizes of one-time pads - birthday paradox matters if the key length is measured in bytes, but not if it's measured in gigabytes, there isn't enough space/time in our universe to expect a collision.
How do you guarantee the third condition? Wouldn't you eventually run into the birthday problem if you sent enough messages?