According to Introduction to Theoretical Linguistics, by John Lyons (Chapter 1), a symbol is a sign whose form is arbitrarily or conventionally associated with its meaning. Also, linguistic signs, namely morphemes, are of symbolic nature. Do symbols, which are signs themselves, include only linguistic signs,morphemes? What I mean is, can symbols be only linguistic units?
First, regarding the question as you frame it in the title, not all symbols are morphemes. Some symbols (linguistic ones) are sounds (which can be "phones" or "phonemes", if you believe in the distinction). Others are structural categories, such as "S", "Determiner", "Stem", "Place". Second, a symbol is conventionally associated to its referent, not its meaning. Saying that a sign is associated with its meaning is an example of homunculus-theorizing (Cartesian representationalism), which adds an extra step in relating symbols and the real world. Rather than saying that "dog" is associated with the meaning DOG and the meaning DOG is associated with the mental concept of dogs, you can simply say that "dog" is associated with a particular mental concept (which in English is realized as "dog").
In many languages (for example Ancient Greek), roots are not words – inflectional material is required to form a word. Roots are morphemes; roots are symbols. Therefore, it is not the case that all symbols are words (thus "no" to "can symbols be only words?"). Symbols do not include just morphemes, they also include phonemes such as the three phonemes (symbols) in /dɔg/. Numerous symbols, such as "stem", are composed of multiple morphemes, so if you're asking whether a symbol is coextensive with the morpheme, it is not.
Nope. A thumbs up is a symbol, but is not a word or a properly linguistic unit.
Consider also Rene Magritte's famous painting "The Treachery of Images". This painting makes salient the fact that a drawing of a pipe is not a pipe itself, but rather symbolic of the pipe.