First of all, good on you for having a go at proper bike maintenance.
Unless you have an expensive American Classic or other 'pro' hub you will be needing 10 3/16 inch balls on either side of that front hub of yours.
Typically these are available from your Local Bike Shop in a pack that will have a few spares for the ones that run away whilst you put them in. The cost of replacement ball bearings is low and, unless you are fiscally challenged, you should always put a new set in when you pack your hub.
The grease that you have is fine, just do not over pack it as this will attract dirt. Cleanliness is crucial, pop off the press-fit washers on the hub body and give everything a thorough clean - GT85 comes in use as does WD40 or even petrol for this application - you want a solvent that will dry off completely and not water down your grease. When you put the press-fit washers back in you will need to use a soft-end hammer or wooden mallet to gently tap them in properly.
The cup and cone flat spanners are needed to put the hub back together again and get pre-load correct. These are affordable tools and not to be used as regular spanners. Normally these are sold with 14/15/16/17mm ends to suit front and rear hubs on most bicycles.
Spindle bearings vary by hub brand so even if the ball count is the same the cone outer diameters can differ. Check the cones and cups are not too pitted - black rough spots - and decide for yourself whether you will be able to live with whatever damage is present.
There should be a gap - as you describe, the gap should be less than a ball's width. Depending on your hub you may need more or less than the expected 10.
Preload is important on the hub. Once upon a time, before this was understood, hubs would be setup so the unloaded wheel would spin 'forever' and stop with the valve at the bottom. However, when you place weight on the wheel there needs to be a bit of extra tightness on the hub to accommodate for that. For me it is a matter of feel although I am sure there are more scientific ways of getting this right. I tighten up the hub so there is no 'play' in it when it is in the fork/frame. Then I give the lock-nuts a proper tighten, in the process introducing extra tightness that makes the spindle feel slightly rough when turning it in your fingers.
When you go on your first ride there might be some grease popping out - wipe this off as soon as you can.
Should you go for adjusting/cleaning the back hub you will need 1/4 inch balls, typically nine per side.
A final note, the hub is usually made by someone other than the rim (and spokes) with affordable-style bicycles. The label on the rim does not identify what the hub is except for with Campagnolo and other posh pre-built wheelsets.