ive never had luck with neons and cardinals, they are not captive bred and because of that tend to be incredibly stressy and delicate little fish by the time they get to the home aquarium...
do NOT mix neons with serpeas, serpeas are well known as fin nippers and tend to be too agressive for neons, and chinese algea eaters can also become incredibly agressive with maturity...
you might want to look into black neons and glowlights if you like the small flashy brightl colored tetras, there not as delicate as neons and cardinals (i LOVE neons and cardinals, but unless your params are perfect, there just very delicate)
i would also suggest adding a small group of corydora and/or ottos corydora are a great bottom dwelling tiny clean up crew, playfull but wont bother your other fish, they like to be in larger groups though so minimum would be 3, and 5 would be better.
ottos are one of THE best algea clean up crews ever, they will eat the kinds of algea most other "algea eaters" wont touch, and they continue to consume algea through their lives, small peacefull and wont bother the other fish (and cute too lol)
shrimp SHOULD be fine with the smaller tetras, the tetra might eat any babies the shrimp produce, but generally should be fine...ive never personally had any of the shrimp, but ive a couple of friends that have kept them very sicessfully.
snails might be another option, but they can breed like crazy...good for algea control though.
given your looking into the smaller tetra you might want to look into adding 1 slightly larger fish, a dwarf gourami would be a nice choice for a 10 gal (only one, there related to betta and need about 10 gals of teritory per gourami) beautiful, generally peacefull with none agressive smaller fish. adding a larger CALM fish will help your tetra feel more relazed and youll see them more.
i personally prefer going fake with the plants, simply because of the types of fish ive kept, id LOVE to do a planted tank one day, but the cichlids would destroy it with digging and the pond fish just eat them lol.
live plants if properly taken care of will NOT foul your water or mess with your ph, infact they work well to keep parameters stable...again in the right set up.
plants tend to need a higher powered full spectrum light to thrive, and even going as far as adding a CO2 pump...
when they thrive they require frequent aquascaping, much like tending bonsai plants...if they start to die however they need to be removed because a deteriorating plant will mess with your water just as leaving a dead fish in there would...decaying matter of any kind isnt good.
so plants would realy depend on what kind of upkeep your planning on doing.
"caves" are always good, untreated teracotta pots and NEW clean untreated PVC pipe make good caves, i personally like terracotta, go to your local craft or garden center and ask the manager if they give a discount for the pots with chipps and cracks ect, i got 8 terracotta pots for my cichlid tank for $1.00 total at joanns because they had cracks in them lol.
i prefer terracotta because they are pourous and will support some algea growth which is great for fish that "graze" (ottos, shrimp, snails ect), i also like the look of the plain red terracotta in the tank compared to pvc.
i think some of the nicest tanks are Natural tanks, pick a focus fish and then base your tank to match that fishs natural envoronment...if you want neons (or most of the tetra) they are
amazon river fish, so do some reserch into natural companions, plants in the area ect.
NEVER mix small tetra, particularly neons with angels...the angel fish's natural diet is the neon tetra LOL!