How old is the battery?
What is the voltage when it is charged up but the engine not running?
It needs to be 12.5 -13.2 v.
Check charging system too.
What is the voltage with the engine running at 4K rpm. You want ~14.5 v.
The engine needs to run above 3-4K rpm for several minutes to allow the stator to recharge the battery after starting the engine.
If you were hearing a single click coming from under the right side cover, that was the starter relay telling you the battery voltage was too low to turn the starter over.
It is a common problem with both cold and hot starts.
Don`t keep trying to start until you drain the battery. Wait 5-10 minutes then try again. It often recovers enough to start.
The Battery Tender Jr only puts out 3/4 to 1 amp I believe, so won`t give much of a boost.
You can boost from any 12 v battery in a car or truck. Most advise to shut the car engine off while while boosting the bike.
The headlight does not come on until the engine starts.
Your`s sounds normal. Once ON, it will stay on until the key is turned OFF.
Is the battery a conventional wet cell with 6 caps on top for adding distilled water?
If so change it out for an MF-AGM like this one:
Thats probably all you will need for reliable starts.
If not, put in some NGK Iridium DPR7EIX-9 spark plugs.
These two change outs cure most starting problems.
If you still need help there are couple of low cost coil mods we can point out to you.
Think about installing a volt meter gauge to monitor the health of your charging system while riding.