Pour a couple of quarts of water into the drain and put one of those thin, circular, rubber drain stoppers on the shower drain. This may keep the trap charged for months. Or just set an old plastic bucket on the drain or invert a bowl on it.
Do not put ordinary auto antifreeze into your drains. The active ingredient is ethylene glycol and it is toxic. There is a practically non-toxic antifreeze using propylene glycol as the active ingredient which is used to winterize plumbing, but even that is unnecessary for your case. Mineral oil might not be good for PVC drains, and is probably not recommended by the city sewage dept, but in small amounts probably would not be harmful to most plumbing and to the sewer treatment processes, but is unnecessary. If you would use it, I would use mineral oil USP from the pharmacy since it is getting into the sewer stream from its use as a laxative.
As far as removing the lavatory, I think code requires one if there is a toilet. Anyway these are an essential element of a bathroom for personal hygiene. Do not under any circumstances remove the existing one without replacing it! You could perhaps install a smaller one if you can find the right one, but this might require a surprisingly costly renovation. If the trap on the lavatory is smelling from sewer gas, then that trap may be drying out too.
Every couple of months (after you pour in some water to charge the trap) pour a half-cup of Clorox into each drain and chase it with a couple of cups of water. this will kill mold in the traps.
Where does your a/c condensate drain right now? It is not a good idea to drain a/c condensate into the drain of a lavatory because it makes an irritating, sleep-disturbing sound.