Is it possible to drive the route from Helsinki at this time of the year?
In general, yes, you should have no problem driving from Helsinki to Saariselkä towards the end of May. Be cautious of moose and reindeer crossing the road in front of you (or lurking at the edges).
How about getting from Saariselkä to the parking lot in Aittajarvi, is it accessible by a car in late-May?
Regarding the specific roads into Aittajärvi itself, these are not maintained in winter. Chances are that in late May you'll be okay, but the best advice before arriving is to contact Saariselkä Customer Service or Tankavaara Visitor Center which can give you the most accurate state of the roads into the park area. Also see the links to historical weather in the middle of this post.
(I couldn't find any photographic evidence of the Aittajärvi parking place in May, but it's roughly comparable (weather-wise) to October, and I found a hiker's blog from October 2014 showing the car park in good order.)
what about the weather and what should we expect?
Naturally, you're better off checking the local weather forecast in some detail before you leave. The most reliable source is the Finnish Meteorological Institute, and although the closest town they forecast for is Inari, it's close enough and the weather patterns are generally stable, so it should be a good indicator. Foreca purports to have a forecast specifically for the UKK national park.
To review historical temperature data, you can look on:
In both cases, you'll see that the average for this month has more recently been a few degrees above zero, and that the projected trend is for increasingly higher temperatures.
Do you think the UKK hike is suitable at this time of the season (if not, does anyone have any other suggestion for a multi-day hike in the Finnish Lapland).
This is the hardest question to answer. Suitability depends very much on your level of fitness, outdoor skills, and hiking experience, as well as the appropriateness of your gear. In general, if you are an experienced hiker and are prepared for bad weather and temperatures potentially around zero, then you should be fine.
I would also recommend not just Lapland, but some of the other hiking areas around Finland. There are some great parks of various sizes, depending on what you are after. Generally, the more rough ground is found in the east of Finland (Repovesi, Koli) but there is interesting nature and scenery to be found all over.
You can check the options for yourself at NationalParks.fi (use the options on the left to filter for national park, national hiking area, wilderness areas, and hiking trail). Remember that there are restrictions on open fires outside of marked areas.
Good luck, have fun, stay safe, and bring plenty of mosquito repellent!