Non e-verify company offering H1

[alekya]

I am in the weirdest situation,and i really need your help to sort it down.

I am in market for 6 months now,(first opt) Just got a full time from a big client who do not participate in e-Verification.They told me,they'll apply H1 this year,but if it doesn't get picked,i need to leave the company,cause for stem opt the company should be e-verified.This is the main clause.My opt ends on june27 2015,need to apply stem then.
My conclusions:
1.Right now,i don't have any offer,so something is better than nothing.(thinking to join the company ...one side)
Other side i got these queries:
1.Since they'll apply in April,results will be in may!!If get picked in H1,How much is the processing time?What would be my status after June 27?Does Capgap need e-verify number?
2.My H1 doesn't get picked up,then I want to change to the e-verifiy company after may and apply for Stem-opt from them.Can i apply STEM in the month of May if opt ends on June 27th? Does this whole senario raise any query?
3.Mainly,in order to apply H1,should the company be e-Verified?

i know its clumsyy,but help me to sort it out.I would really appreciate, if you people give me some clarity in this situation.

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[DesiOPT-FaceBook-Admin]

Your message was posted as anonymous on Facebook wall- (Dec 7th)

[DesiOPT-FaceBook-Fan]

1 nope cap gap does not need everify
2 yes u can but be care full with timing .

[DesiOPT-FaceBook-Fan]

If your h1 get picked in lottery you will be eligible for cap gap ! You can apply for opt extn any time before your opt expires but you will not be eligible to work after 180 days after your opt expires !!