It is a typical construction for an indirect statement. “The doctor says THAT Quitus is breathing and his heart is beating.” It’s still a single sentence, the subject and predicate being Medicus dicit and the direct object is actually what the doctor said–the entire indirect quote. Note that it would look a lot different were it a direct quote: Medicus dixit “Quintus spirat et cor eius palpitat.”
Since you’ve got Wheelock’s, check out chapter 25’s discussion of the infinitive with indirect statement with accusative subject (p. 164-6 in the latest edition). They explain it much better than I.
134 Aemilia non putat medicum puerum aegrum sanare posse.
I read this to mean: Aemilia does not think the doctor is able to heal the hurt boy. Why is posse in the infinitive?
Once again, it’s the same type of construction as with the indirect quote: “Aemilia does not think THAT the doctor is able to cure her sick boy.” The subject is Aemilia, the predicate non putat, and the object the remainder of the sentence, or, in this case, what Aemilia is thinking. Once again, check the Wheelock site listed above for a highly detailed explanation.
BTW, I am neglecting my study of Wheelock’s for Lingua Latina. That can’t be good…can it? I am able to read through and understand pretty well. Simpler sentences I am able to read without translating to English. This is thrilling.
Personally, I use Wheelock and M&F to supplement what I learn in LL. So, if there is a construction that isn’t familiar to me or doesn’t “click” in my mind while reading about good ol’ Marcus and Quintus, I look it up and read the more thorough explanation in my reference grammars. I don’t hammer out the exercises in either anymore, though there was a time when I did all of the ones in M&F. I do not, however, let myself proceed to the next chapter in LL without making sure that I understand the grammar.
Should I worry that I could not produce the Latin as well?
It’s all up to you. If you’re learning on your own, you may have no desire to compose Latin…you may just want to read the Classics in their original language, in which case an emphasis on writing isn’t entirely necessary. If you wish to be able to write Latin prose, poetry, etc., it’s best to acclimate yourself to it as you go through LL. Spend time with PENSVM C in each chapter and write out each answer fully. It’s a good start.
I read over the Pensa, but I can’t check my answers (are these online anywhere?)
The answers can be found in the LL Teacher’s Manual. It contains the answers to all pensa, in both LL I and II, as well as the answers to all of the additional exercises found in the supplemental Exercitae. It’s available on Amazon, through Pullins, and I’m certain your local bookstore could order it for you.
I have a nagging feeling about pushing on, but then I think, What could be the harm of it? Wheelock’s isn’t going anywhere. Am I just too stuck on this way of learning Latin from my (much beloved) Latin professor from college? I am wondering this now from reading the recent posts on frustration with Latin.
There is certainly no harm in just proceeding according to your enjoyment, if you’re not in any class. In fact, if you’re not enjoying it, you are almost guaranteed to give it up eventually. Leisurely go through LL if that’s what you want to do…you’ll learn a ton and can fill in the blanks (if any exist) later if you desire. Learning a language is a tough enterprise for some of us…I know it is for me. My beloved Latin teachers all had differing methods…one even used a textbook written in 1659. However, even the textbook written in 1659 uses a teaching method very similar to that of LL…tried and true…it’s worked for almost 400 years…you can’t argue against that. 
Best,
Chris
::I edited this to remove what Amadeus better explained ::
I didn’t recognize what I was seeing in those sentences - couldn’t put it together.
but she did give us copies of copies of copies…