Here is chapter and verse of the regs for your eyes on the First Class medical:
§67.103 Eye.
Eye standards for a first-class airman medical certificate are:
(a) Distant visual acuity of 20/20 or better in each eye separately, with or without corrective lenses. If corrective lenses (spectacles or contact lenses) are necessary for 20/20 vision, the person may be eligible only on the condition that corrective lenses are worn while exercising the privileges of an airman certificate.
(b) Near vision of 20/40 or better, Snellen equivalent, at 16 inches in each eye separately, with or without corrective lenses. If age 50 or older, near vision of 20/40 or better, Snellen equivalent, at both 16 inches and 32 inches in each eye separately, with or without corrective lenses.
(c) Ability to perceive those colors necessary for the safe performance of airman duties.
(d) Normal fields of vision.
(e) No acute or chronic pathological condition of either eye or adnexa that interferes with the proper function of an eye, that may reasonably be expected to progress to that degree, or that may reasonably be expected to be aggravated by flying.
(f) Bifoveal fixation and vergence-phoria relationship sufficient to prevent a break in fusion under conditions that may reasonably be expected to occur in performing airman duties. Tests for the factors named in this paragraph are not required except for persons found to have more than 1 prism diopter of hyperphoria, 6 prism diopters of esophoria, or 6 prism diopters of exophoria. If any of these values are exceeded, the Federal Air Surgeon may require the person to be examined by a qualified eye specialist to determine if there is bifoveal fixation and an adequate vergence-phoria relationship. However, if otherwise eligible, the person is issued a medical certificate pending the results of the examination.
https://www.ecfr.gov/cgi-bin/text-idx?c=ecfr&sid=5c0d1766e37c2027601b6684dcd4e1bd&rgn=div8&vie w=text&node=14:2.0.1.1.5.2.1.2&idno=14
Also, from the ever useful Guide for Aviation Medical Examiners:
https://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/ame/guide/app_process/exam_tech/item31/
I don't think there is any test for depth perception on the regular FAA Class I exam. There is a test for exophoria and esophoria, whether your eyes want to turn inward or outward too much when fusing the left and right images together. On the common Titmus testers, you are given a line in one eye and musical note symbols in the other and you are asked where the line meets the notes. On all of the testers I've encountered over the years, if you pick a note near the middle, you'll be OK.
Most of the testers seem to use slides similar to these (some of the answers seem to be the same for decades):
http://d13iwp969lq78m.cloudfront.net/pdf/slide-packages/Ophthalmic%20Prescreen.pdf
Remember, even though you are the patient when you go to the Aviation Medical Examiner, like a check airman, he or she is working for the feds so any information about your eye problems that you confide is fair game for documentation. Admit nothing, deny everything, and make counter-accusations is my advice. 
In other words, don't volunteer the fact that you don't have depth perception to an AME or a company doctor if you don't have to.