Wow, you have many misconceptions and some bad info. All you have to do is Google replace chain to get all the info you need (look at written instructions 1st, videos only for additional support) but here's a summary of the points you brought up.

• Whatever "people" mentioned a 9 speed chain were flat-out wrong. If you have 10 total speeds (2/5, 2/6) then the rear is a 5 or 6 speed. As already noted a few times above, the designation on chains goes according to the rear cogs, not total speeds of the bike. Five rear cogs was long ago obsolete, so many chains don't even mention that - most say 6-7, 6-8, 7-8, etc. DO NOT go higher than an 8 speed, and better to get one that specifies 6 as part of the designation. A narrow chain may well shift more poorly rather than better. In any case you don't need a solution for shifting when the problem is not getting to the small cog (which is HIGH gear, not low) but rather skipping in that gear.
• The "someone" who explained about "stretch" (really just wear) is correct in some ways, wrong in others. The old chain is worn, and under the higher pressure present when in the highest gear the chain wear shows up as skipping. A new chain may skip on an old rear cluster due to some of the cogs being worn (typically the smaller ones). You need a new chain, but "they" were wrong about a 9 speed chain being the solution.
• The chains designed for higher cog counts are narrower, not shorter. If you do the work yourself you need to pay attention to proper chain length sizing instructions in any replacement tutorials.
  Again, Google replace chain - sheldonbrown.com and parktool.com are recognized as the best sources. You absolutely do not need a "chain checker" as the symptoms clearly point to chain wear, but if you did I would recommend a ruler instead. You will need a chain tool.

p.s. Although the skipping in the small cog indicates the chain is worn it is also very possible that you are pushing too high a gear for conditions. The high gear on a 10/12 speed would be appropriate for speeds upward of 25mph, and not under acceleration, but rather to maintain that speed - which most riders cannot do for very long other than downhill.

You did this when?? For a previous chain or have you installed a new one? Newer narrow chains should not be joined by reusing the old pin. Not only are they more subject to failure but with the wide availability and convenience of connecting links there's no point. Google is your friend again here - reuse chain pin.

I would say that it's a bit odd seeing from past posts that you have been reselling bikes that you have "gone over completely." I don't see how you can own that many bikes and be selling to others when your knowledge is so lacking about something as critical as a chain.