Before answering the question; I need to become clear about the concept of common ancestry.
All groups of organisms seem to have only 1 common ancestor (earliest ancestor of all life-forms on Earth), as generally accepted.
While discussing about evolution; when people frequently talk about "common ancestry";
firstly, they talk about latest common ancestry. If we pick any 2 species without any basis (for-say a pea plant and a dog); and try to search back their latest common-ancestor; there should exist some-or-some life-form (usually extinct, commonly hypothetical).
secondly; that latest common ancestry too; would must 'exist' between any 2 randomly chosen life-forms. Latest common ancestry is of not much use if we think it in a present-or-absent way. It only makes sense when we use it in a quantitative way ("how much common or distant") within 1 particular lineage-of-interest.
Example: in the following sample
* phylogenetic tree;
* drawn on basis of this, this and this)
People often informally say "Macaca mulatta monkey and human have a latest common ancestor" or "cats and human have a latest common ancestor" but that would not make any useful sense. But it will make a proper sense only if we say "monkeys share more close common ancestry with a human than cats" or "cats share more close common ancestry with a human than plants". or more elaborately; "cats share more closer-distance of common ancestry with human than the distance of common ancestry cats share with plants."
An answer to your main question(s);
- We can say neither of "cats evolved from monkeys" or "monkeys evolved from cats" in a true sense. They evolved from a certain group of organisms (their latest common ancestor) that doesn't exist today.
Here is a diagram from http://genomewiki.ucsc.edu/ (slightly modified by marking the cat and monkey)
( original file, page)
(Rhesus macaque is common-name of Macaca mulatta monkey)
The website http://evolution.berkeley.edu/ has wonderfully explained, evolution proceeds like a tree, not like a ladder.
Still, we can argue that, at the branching-point, out of 2 evolving groups, only one evolving group could have shown alteration and another group was just unaltered parent group, like this;
Still, we could not apply that for cat and monkey; because if we consider one of the 2 branches after red circle at fig. 2 (for argument say it is on cat's side) exact same as parental-species; we could not say "monkeys had arrived from the cat". That ancestor could neither be considered as a cat or cow or horse etc.
- Cats and monkeys have enough similarities. Each of both contains 2 eyes, 4 limbs, 1 tongue, skin hairs, ear-pinnae and such and such. Is it too difficult to recognize?
Yes, we have mainly 2 tools to recognize common ancestry...
A. similarity (including macroscopic through biochemical to molecular-genetic);
B. Fossil records, which are much difficult to find (mainly structural, also ancient-DNA evidence nowadays doing great help
And yes, establishing common ancestry from similarity, is difficult, because there are lots of exceptions caused by 'convergent evolution'. But still, there is a strong correlation between similarity and closeness of ancestry; and it needs our discretion about which one similarity could be really due to same origin, or which one is due to convergent evolution or due to just a superficial similarity or coincidence.
Understanding Evolution website: http://evolution.berkeley.edu/ and their section on understanding evolutionary tree (http://evolution.berkeley.edu/evolibrary/article/evo_05).
** The Evolution of Plants by K. J. Willis and J. C. Mc.Elwain/ Oxford: that includes an elaborate discussion about ancient DNA in animals too.