Of Las Vegas airports, Henderson is certainly the simplest to get into because you can approach from S without entering the class 'B' airspace. However it's a long way from the strip and the famous free shuttle is ancient history. They do have car rental on site though.
McCarran certainly accepts VFR traffic but 'handling' is compulsory and seems expensive, until you go to pay for the fuel! That was $7 per gallon in January and doubtless more now. Afterwards, the 'handling' seems cheap by comparison! Mc Carran is quite demanding for the VFR pilot but ATC are very patient.
North Las Vegas is very accomodating, but also bereft of a free shuttle in these leaner, meaner times. Reaching it means a transit of the class 'B' and you would definitely want a briefing on this prior to going there, but actually it's not that difficult. You can get to N las Vegas from the NW without entering the 'B' but it's a long way round. Be sure to prearrange car rental because the desk is not attended.
The advice about July and the temperature is very good and the solution is to fly early as others have said. Most places are open at 6am and it helps to stay close to European time anyway. After about 10:00 it gets very bumpy indeed. Don't expect to fly 4 up in a 172 out there: one passenger is enough. The Las Vegas airports are all around 2000' elevation so it's all about density altitude.
The Las Vegas 'B' can be intimidating and an alternative could be the Laughlin/Bullhead City airport in uncontrolled airspace and very scenic down by the river. They have free hotel shuttles. While there you can also visit Lake Havasu which is pleasant enough with a superb FBO - Desert Skies.
I dont really agree about the scenery - to a European the trip up there is highly scenic especially once you escape the murky LA visibility. I'd consider doing the trip in two early morning segments with a stopover in Victorville or Barstow (though niether place has anything to reccomend it) so that you arrive in the Las Vegas area feeling fresh.
I started my US flying career out of N Las Vegas with a lot less experience than you and still go back there whenever I can. I never grow tired of the amazing scenery and the relaxed, straight forward flying. Enjoy!