During Linnaeus' lifetime there was two ways to get in contact with exotic specimens other than travelling abroad: by visiting museums and collections or by receiving the specimens at your home/laboratory✻. Natural history museums were not that popular at Linnaeus time but only some time later, in the nineteenth century (the first natural history museum, the Muséum National d'Histoire Naturelle, was created in Paris in 1635).
That leaves us (or even better, Linnaeus) with the second option: receiving the specimens. According to the University of California, Berkeley:
Linnaeus continued to revise his Systema Naturae, which grew from a slim pamphlet to a multivolume work, as his concepts were modified and as more and more plant and animal specimens were sent to him from every corner of the globe. (emphasis mine)
You could argue that a (dead) llama is a big animal, but one can describe an animal using just its bones, or even just some of its bones. Besides that, it's worth mentioning that Owen received a whole mammoth in UK to study and describe! So, sometimes, size is not a problem.
Answering your second and third questions, a good deal of the animals he described he never actually saw in the wild or alive. Estimating the precise fraction is way more complicated. The same may be said regarding plants, but I guess that the fraction here is smaller: live plants are easier to maintain, seeds are easy to be transported and sown etc. Besides that, botany was Linnaeus' area of expertise. The same link above says...
Linnaeus was also deeply involved with ways to make the Swedish economy more self-sufficient and less dependent on foreign trade, either by acclimatizing valuable plants to grow in Sweden, or by finding native substitutes. Unfortunately, Linnaeus's attempts to grow cacao, coffee, tea, bananas, rice, and mulberries proved unsuccessful in Sweden's cold climate. His attempts to boost the economy (and to prevent the famines that still struck Sweden at the time) by finding native Swedish plants that could be used as tea, coffee, flour, and fodder were also not generally successful.
... which shows us that he dealt with live exotic plants.
Regarding your fourth question ("did he name species based on other people description but without ever seeing one dead or alive?"), the answer is yes, specially for plants and insects: there are several specimens that Linnaeus described based just on drawings or paintings (which may be considered other people's description), without ever seeing the actual organism. Two examples are Dysdercus andreae (an animal) and Porella (a plant).
PS: Not related to this question or to biology, but this is (unfortunately) quite common in sciences: some of Freud's patients, among them his most famous cases (Anna O., little Hans, etc...), he never actually met or even saw at the distance! He "diagnosed" them using just descriptions and hearsay...
✻ According to RHA comment below, there is a third way: sending people to collect the specimens for you.
