You aren't getting a conventional loan unless you have several years as a 1099 of similar income.

As a foreigner getting conventional lending might be an obstacle and I think there will be many people on this forum that can provide resources to guide you through. 

This is pretty low hanging fruit: If you do not already own real estate (personal occupancy) I would start by purchasing a SFH or 1-4 unit using a conventional mortgage:1.

For conventional loans via Fannie, I know that there is a 60 day seasoning for downpayment funds for conventional loan. 

Try buying a MF in E Clev, good luck there is no inventory for a reason :)  Again I am going by my 10 years of doing 100s and 100s of deals there, 1ks or reno/repairs, 1ks move in and outs working with all the govt agencies.

Would there be issues with the 30-year conventional bank loan itself for the 12-unit apartment since the down payment is from a HELOC?

I would imagine a conventional loan would take too long to get approved in order to close before the foreclosure auction date?

Could I get the usual conventional loan?

You will not get anywhere near that in SH, or CH, BTW good luck finding a MF, no inventory.

Cheyenne,You should attend the Noteworthy convention in Las Vegas this October where you'll have a chance to meet with quite a few people in the note and real estate business.