If you go to Export > Quicktime movie
Then in the settings tab - leave as (or change to) ''current settings''
This will export the timeline in the same settings you have been editing in, which is most likely the same format you captured the footage; unless you have changed the sequence settings somewhere along the line.
This will give you an export in the best quality. It does however leave you with a finalcut .mov file which isn't the most versatile file type if you're viewing this on anything other than a mac with final cut. BUT you can easily convert it from here in to any format you like
If you do want to export it in the same format as you have tried but without interlacing you should try :
As you did before go to export > Using quicktime conversion > click the ''advanced'' tab next to ''settings''
Then click the ''size'' tab and you will see an option to check ''de-interlace source footage''
This will not always resolve interlacing issues but is the first port of call if you're trying to figure it out.
I hope this made some kinda of sense enough to help