The standard figured bass is usually just a 7 for a 7th chord in root position, right? If this is correct, the 5th and 3rd are assumed. But what about if the 7th chord should omit the 5th and double the root?
Traditionally, this isn't specified. A single "7" in the figures could be performed either complete (with the chordal fifth) or incomplete (without).
Rarely, though, the figures can specify particular doublings above the bass. For instance, in a four-part setting, you might occasionally see "8/7/3" in the figures, suggesting the bass is doubled (that's the 8) with no fifth above, just a seventh and third. But as I said, this is an exception to the rule; musicians are rarely this specific with figured bass.
It really depends on the context since there is no standard way to denote this. In a harmony class, you should ask your teacher which kind of notation they would find acceptable for the omission of the fifth.
In a baroque piece, impossibly broadly speaking (baroque covers all the styles and genres encountered in over a century of music) only the opposite is possible: requiring the fifth to be there by writing in a 5 along with the 7. Only a 7 means the fifth may be omitted (and rather often is). (Of course, if both 3-7 and 5-7 appear regularly in a piece without any other apparent reason, 3-7 is probably without 5. Also in certain contexts the third can be omitted and not the fifth.) There are some situations for which primary sources specify rules, such as playing sequences of sevenths resolving to sixths on a stepwise descending bass being without a fifth (of course as with any rule there are exceptions).
Fortunately there is another factor beyond personal (but presumably stylistically informed) taste in the decision of whether the fifth should be played: voice leading. For example, a series of seventh chords on bass notes alternately falling by fifths and rising by fourths has a very natural voice leading if each second chord is without fifth, but becomes awkward when the fifth has to be present in each chord. Of course, this factor is strongest only in styles which require strict voice leading (such as German 4-voiced continuo); other styles (e.g. Italian continuo) are freer contrapuntally but include ideas of (a.o.) dynamics and character to guide this type of decision.
In many baroque styles however, numbering is very sparse (since the performer is expected to be able to make their own numbers, depending on the style either in advance or on the spot) so there was no real need to notate something as specific to the realisation as the omission of the fifth: any reason a composer might have to object to adding a fifth to a seventh chord should also occur to a skilled continuo player.
Post baroque, we have galant continuo and per the book of CPE Bach (yes, son of) that is, as far as it is possible to condense a whole book into one sentence, very often in three voices anyway (so there is no voice left for the fifth, but all rules in CPE’s book come with labyrinths of exceptions).
Before the baroque era there is some limited continuo too but in those cases the continuo generally plays what the melodic parts play minus diminutions, so whether the fifth should be added is decided by whether it is present in the other parts.
Chord notation systems, including figured bass, leave room for interpretation. It lets the performer or the arranger to interpret the composition their own way. In order to specify the exact notes to be played, perhaps it's better to write them down, rather than use chord notation.
Figured bass notation indicates what notes are supposed to be played. Everything else is interpretation, following style and conventions. In particular there is no way to indicate that a given note should not be played.