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I would like to know which one of the following is correct:

1. Figures 1 and 2 indicate that...


2. Figure 1 and 2 indicate that...


3. Figure 1 and 2 indicates that...


4. other possibilities?

Try substituting your terms for others while keeping the grammar, and it'll be a bit more natural.

1. Cats Bob and Jim cough up hair balls

2. Cat Bob and Jim cough up hair balls

3. Cat Bob and Jim coughs up hair balls

I'm pretty sure you should be using the plural form (Figures, Cats) for the noun. I'm not so sure about the verb (indicate, cough), but using the singular seems to imply that it's something that happens in general, while the plural is more active. I'd go with 1 myself.

The correct is
Figure 1 and 2 indicate that

We do not use "Figures" because the word is not used as common noun but as a proper noun with identifier 1 and 2.
So it is basically "Figure 1 and Figure 2"; to reduce wordings we compact it as "Figure 1 and 2". Capitalization of "F" is required.

Since subject is plural the verb would be "indicate".

Though, some journals accept and prefer as Figures 1 and 2, so in such case, we edit as "Figures 1 and 2 indicate". "Figures 1-4 indicate"

Best Sci-Edit Publications (sci-edit.net)

I agree with Sci-Edit Publications. Bob and Jim (as per xpdite's examples) are names/titles and not positions (e.g., 1 and 2), hence their usage is different. Consider the collections in the attached image. You can refer to an item in a collection either by its name/title or its position. The names/titles of figures in a text could be lengthy or identical, hence are preferably identified by their position.

In my opinion, the following is correct, depending on the intended usage:

1. 
   Bob and Jim cough up hair...


2. 
   Cat 1 and Cat 2 cough up hair...


3. 
   Cat 1 and 2 cough up hair... (short form)

Similarly:

1. The age chart and population plot indicate that...


2. 
   Figure 1 and Figure 2 indicate that...


3. 
   Figure 1 and 2 indicate that... (short form)

Notice the use of "Cat" and "Figure" instead of "cat" and "figure".
And if this is still unconvincing, consider the sentence: January 1 and 2 are public holidays.

If you aren't writing for Sci-Edit Publications, you may be interested in how other publications handle in-text references to multiple numbered figures (illustrations). As a freelance copy editor, I have worked for many book publishers with differing preferences on such things as whether to capitalize the word figure in an in-text cross reference, whether to identify individual figures within a chapter as (for example) Figure 1-1 or Figure 1.1, and whether to render the word figure in the caption itself as initial-cap F and lowercase igure or as large-cap F and small-cap IGURE.

In a similar vein, some publications insist on repeating the word in in-text references to multiple figures (as in "see Figure 2-6 and Figure 2-7"), while others approve of using plural figures in such situations (as in "see figures 12 and 13"). But I have never worked for a publisher that insisted on using singular Figure as a lead-in to a reference to multiple figure numbers (as in "see Figure 4.6 and 4.7").

The Chicago Manual of Style, sixteenth edition (2010) has a fairly brief discussion of in-text cross references, which includes these relevant remarks:

3.9 Text references in numbered illustrations. If there are more than a handful of illustrations in a work, they normally beat numbers [cross reference to a discussion of working numbers for unnumbered illustrations omitted], and all text references to them should be by the numbers: "as figure 1 shows...," "compare figures 4 and 5." ... In text, the word figure is typically set roman, lowercased, and spelled out except in parenthetical references ("fig. 10").

Evidently, Chicago is not swayed by the theory that "Figure 1" is a proper name, rather than a simple number-associated designation.

Most (but not all) of the U.S. publishers I've worked with follow Chicago in lowercasing the word figure in cross references. On the other hand, many (and perhaps most) break with Chicago in spelling out figure even in parenthetical references. I should note that most of my work is in mainstream nonfiction and college textbook publishing, rather than in strictly scientific publishing, that work experience may skew my impression of industry standards somewhat.

If you are writing or editing for a mainstream U.S. publisher, your primary authority for handling in-text references will be the publisher's house style. If it doesn't have a rule on this point (which is unlikely) and your text isn't governed by a specialty style such as MLA, you are free to act on your own preferences. On this point of style, as on most others, Chicago's guidelines are generally reasonable and consistent, and they are widely respected in U.S. publishing. I would be more inclined to follow its recommendations than those of a random publisher that happens to be very enthusiastic—not to say proselytical—about the idiosyncratic house style that it has adopted.

To sum up, if you choose to follow Chicago to the letter, you would express your example as

Figures 1 and 2 indicate that...

or

Figure 1 and figure 2 indicate that...

On the other hand, if you establish a rule to spell Figure (when used to indicate numbered illustrations) with an initial-cap F throughout your book, but elect to follow Chicago in all other respects, you would express you example as

Figures 1 and 2 indicate that...

or

Figure 1 and Figure 2 indicate that...

Either way, the figures are plural and the verb indicate is plural, too.