Does the use of "var", "x", and/or "ssp" in a scientific name provide specific information?
Yes, those terms denote the rank of the taxon in question.
Does the use of " var " mean that the plant is a variety?
Yes, the use of "variety" or "var." in a taxon name indicates it has the rank of variety. There is also "subspecies", "subsp.", or "ssp." to indicate the rank of subspecies.1
Does [the use of "var."] mean that each of these [taxa] have been cultivated and/or bred by a human?
No. Subspecies and varieties are wild taxa found in nature (mostly, some common crops are called species, such as Zea mays subsp. mays).
Does "ssp" mean that the plant is distinct yet compatible with others in the species?
Probably. This is complicated because it gets into species concepts, but some kind of distinctiveness is required, and the biological species concept (where a species is basically defined as being able to interbreed) is popular.
Does the x indicate that a plant is a hybrid, and are the varied uses indicate a different meaning?
Yes. The multiplication sign × is used to indicate a taxon of hybrid origin. Its different placement can indicate whether the taxon is an intergeneric or interspecific cross. If the × glyph is not available, a lower case x is used (though, really, this should be rare these days). The × should occur before either the genus name or before the specific epithet (e.g., the species Populus maximowiczii is just a regular species name that happens to contain the letter "x").
Is there a way to identify a that a plant is cultivated based on its scientific name?
Yes. Names that include quotes (specifically single quotes), non-italicized text, or the addition sign + could all be indications that the plant is a named cultivar (or group, grex, or graft-chimaera).
Since the question references the USDA plants database, I'll limit the answer to plants. Similar rules are applied to animals and microorganisms.
Each entity being referenced is a taxon (plural taxa), that is, a group of populations of organisms that are seen by taxonomists as forming a unit. Taxa are hierarchically ranked, with the highest being kingdom2, such as Plantae (the green plants), the "basic" rank being species, such as Populus maximowiczii, and—most relevant here—ranks below species, such as subspecies and variety.
The rules for how (wild) plant taxa are named are defined by the International Code of Nomenclature for algae, fungi, and plants (the Code). There are similar codes for animals, prokaryotes, viruses, and, interestingly, cultivated plants.
While the Code recognizes that there are an indefinite number of ranks, it gives names for many of them from kingdom down to forma (grouping them as principle or secondary ranks). It also specifies that more ranks can be created by adding a "super" or "sub" prefix to the rank name to indicate a higher or lower level rank, respectively.
Species is the lowest principle rank, subspecies is below species, and variety is below subspecies (and, for completeness, there is forma below variety).
Strictly, since subspecies and variety are different ranks, a taxon could be classified as a variety that itself is a part of a subspecies. In practice this is rare, usually a taxon is assigned to one or the other.3
Besides denoting a different rank, what is the actual difference between a subspecies and variety? It turns out that the biological definition of these terms is basically up to the taxonomist doing the work. Just like there's no set definition of species, there is no set definition for intraspecific taxa. Some taxonomists might basically rank all intraspecific taxa as subspecies or as varieties, depending on their philosophy. Those that use both terms need some recognition that variety is a narrower term than subspecies, but what that looks like operationally is up to the person/group doing the work.4
(However, one should follow the resource they are using. If it's listed as variety in USDA, use variety. It's incorrect to just decide you don't like it and change it to subspecies [for example]; that would require a research publication.)
Species, either members of the same genus or of different genera can sometimes interbreed and form a natural population. One could refer to these by simply listing the hybrid parents in a hybrid formula, such as Agrostis stolonifera L. × Polypogon monspeliensis (L.) Desf.5 One could also formally name the resulting hybrid. In this case the hybrid nature would be noted by placing the × before either the genus or species name. The above intergeneric cross between Agrostis and Polypogon has been given the name ×Agropogon lutosus C.E.Hubb. The name Populus ×canadensis refers to the interspecific cross between P. nigra and P. deltoides.6
The basic unit of cultivated plants is the cultivar, which is an assemblage of plants selected (by humans) for a specific character, and that remains stable through appropriate propagation. The naming of cultivars is governed by its own code. This code only recognizes the terms subspecies, variety, or form, as they apply in the (wild) plant Code. (Note that other groups, such as the Association of Official Seed Certifying Agencies, do consider cultivar and variety to be synonyms.)
An example cultivar name is Clematis alpina 'Ruby'. Note the use of the correct botanical name, the cultivar epithet in plain text, and the use of single quotes. See other examples here.
1The Code recommends the abbreviation "subsp." for subspecies, but "ssp." is very commonly used.
2Kingdom is the highest rank recognized by the Code; there are generally recognized ranks that are higher, such as superkingdom and domain.
3For example, take the taxon Saxifraga aizoon var. aizoon subvar. brevifolia f. multicaulis subf. surculosa Engl. & Irmsch. This has been fully classified to a particular variety, subvariety, forma, and subforma. To refer to this taxon, one would almost always just write Saxifraga aizoon subf. surculosa, and that is in fact it's name as defined by the Code. One would only spell the whole thing out if they needed to give the full classification. Since no other plant can have the same name, there can be no other subforma within Saxifraga aizoon with the epithet surculosa, even if it were to belong to another variety, forma, etc.
4I could include forma in this paragraph, too, but this seems to have greatly fallen out of fashion as a useful rank.
5I've also included the author abbreviations here. They have their own set of rules. Just know they're not technically part of the name, but often included to help avoid ambiguity.
6This could be written as Populus ×canadensis or Populus × canadensis. The × glyph is not actually part of the name, and can be placed for best readability.