The best way to come in is via the South: Bulgaria. You could follow the Danube river and cross by the barge. Taking the bridge (gates of steel) is nothing interesting.
It is really nice. Be careful that if you plan to stay in hotels there is nothing on the romanian side for at least 150km heading North. You'll see vast empty expenses and the roads are good.

As far as the mountain themselves, best not to do the Transalpina over the weekend. It is packed... There are two parallel roads going up the mountain. You could do both, but that is a day in itself.

A super nice place is Sigisohara. Medieaval well preserved city. Pricey hotels but really worth a look. All of the area in the Carpathian present is really great on a bike. I went at least 3 or 4 times and enjoyed it very much. The roads are mostly ok.

For sleeping, if you camp, stay away from gypsy villages is my advise. If you sleep in hotels, there are plenty in most medium sized cities: use the OSN maps or the garmin maps and look for places. I never failed to find a proper place at the 3rd attempt max. I look for places where I can park the bike safely and never had a problem.

I really advise not going via Hungary: it is a boring straight ride. On the other hand coming via Bulgaria gives you the opportunity to ride thru the Balkans which is a lot better.

If time allows, and you any more challenging rides and some off-road, you could cross into Ukraine and go into the Ukrainian Carpathian. Very hard roads with potholes the size of a 747 and murderous if it happens to rain (you don't see the depth of the holes). But incredibly scenic.

As far as safety and food, only good thing to say! Just be cautious where you park your bike for the night and all is fine.

Anyway, let me know if you need any more information.