EricR, welcome! You don't need to enter the lottery, because there will be plenty of permits available during May. Here are the unused permit numbers for 2014, with additional data added: Unused Whitney Permits 2014
. Scroll up to the top of that thread to see links to earlier and later years. (Reminds me I need to add the unused 2017 numbers).
If the snow does not pile up between now and May, you will still have long sections of snow completely covering the trail. You will need the ice axe because the slope is that steep. If you have no experience, you need to learn how to self-arrest and also watch the ice axe videos. (See the Reference forum here, first post)
You sound pretty inexperienced. If you can change your plans at all, then I would plan on coming out mid-week the last week in June. Even then, there are always plenty of walk-in permits, and you will have a far greater likely chance of hiking a better trail.
Also, check through the Orientation Notes (link above left). You need to know about AMS -- altitude sickness, in case it affects you.
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As for alternate hikes, any of the roads on the East Side (of the Sierra) have trails from the end of the road up to the Sierra Crest. Kearsarge Pass out of Onion Valley (town of Independence) is the easiest and shortest to the 11k pass leading to the drainage to the west. It, too, will have lots of snow the first of May. Most hiking doesn't really start here until mid-June, unless you are happy to tramp over lots of snow.
If the road to Horseshoe Meadows is clear (left turn off Whitney Portal road), it is also a great place to see -- HM is a high plateau at 10K, with trails leading to Mt Langley, the Sierra's southern-most fourteener.