I am told that I cannot double the 3rd when constructing a 4 part triad but at the same time I am told that the only note that can be omitted is the 5th. So in a triad, if I omit the 5th, then all I have left are the 3rd and tonic to double. Is this a bit confusing to anyone else but me?
I have a slightly different viewpoint as other answers: no, I don't think that part-writing rules are contradictory. But unfortunately, some teachers (and pedagogical approaches) are contradictory with the guidelines that they create.
I say this for one simple reason: there is nothing wrong with doubling the third of a major chord. Composers do it constantly on I and IV in major keys. The only time it's incorrect to double the chordal third is in the V chord. But this isn't an error with doubling the third of a major triad, it's an error of doubling the leading tone.
So let your doubled thirds of major chords fly; they're not errors, and if someone says they are, just point them to one of thousands of examples where composers had no problems doing so.
Lastly, there are guidelines of different weights, meaning that some of these possible part-writing transgressions are more serious than others. So when you're stuck in a situation where you have to decide which guideline to ignore, you'll of course want to go with the lesser problem. In a situation where you'll have either "hidden" octaves or parallel perfect fifths, go with the route that has "hidden" octaves; it's a far less serious transgression that parallel perfect fifths.
In Walter Piston's Harmony he often says melodic considerations can override harmonic ones.
From that perspective you might have an occasion where doubling a third would be acceptable.
The point isn't to then try quantifying, or making a long list of exceptions for, acceptable doublings. You want to understand the reason for the norm in the first place, and then understand such "rules" aren't written in stone.
I think Piston's explanation about doublings is pretty easy to understand and not contradictory. You either double chord roots or you double the tonal scale degrees. The reasons are to keep the identities of chords and the tonality clear.
Also, I'd like to reiterate what @Richard said in his answer, it's common to double the third of a major chord. The cases I'm thinking of aren't necessarily four part harmony, but I don't think it matters. I'm thinking of classical, solo piano music. You often find a tonic major chord voiced as root and third in the accompaniment with the third doubled in the treble melody.
In terms of chord and tonal clarity this isn't a problem, because the surrounding harmony will usually be very much tonic/dominant and there will be no mistaking the tonal center. Melodically the treble plays the third in what you might call a "pastoral" style where the melody pivots around the third for a soft and gentle sound. You could say the accompaniment should just play the tonic alone to avoid a doubled third, but that might sound a bit thin. You could play root and fifth in the accompaniment and third in the treble for a balanced three part triad, but the root/fifth in the accompaniment sounds a bit like old Baroque voicing. Root/third accompaniment sounds rich and seems to have appealed to classical composers. So both parts separately do what they need to. The doubled third of the combined parts simply isn't deemed a problem.
I don't mean to belabor the point. I just wanted to give a real example of how parts are handled in a way that may seem contradictory to the fundamentals of four part harmony. It's probably best to consider those four part harmony rules as norms for vocal music (or strings only) and your exercises demonstrate you can control part writing to conform to the rules. Beyond that you will find varying norms depending on style and instrumental considerations.
I don't know if the rules are contradictory but they are more rules in the sense of what sounds good, to the culture the music is created in. But everything involves compromise and it can be difficult coming to a solution. I read one author who said that four part harmony rules were created by 19th century theorists retrospectively.
When learning harmony or counterpoint it is better to follow the rules - as they are called - especially in a test or exam. This means it sometimes takes a lot of work to write something that is compliant with the stated rules.
Doubling the 3rd in a major chord should usually be avoided, in a minor chord you can sometimes get away with it. I personally do not like omitting the 5th so would probably work to avoid that. Although not usual, I once spent many, many hours over a couple of days writing eight bars of counterpoint. It doesn't usually take me that long but the important thing is to get it right.
The other thing is to consider the source of such information, they may be a professor of counterpoint, or they may be someone who knows less than you.