Joe,
I'm not sure where you are located but it definitely sounds like you need a gym with better equipment (Really?? playground rings?!) - and a program that has a coach that is familiar with the USAG competitive level routines.
At age 15 - practicing one hour a week you are not at all likely to succeed (why has someone not shared this?! I don't want to be the bearer of bad news - but blunt honesty is needed here.) on on levels (Level 8, 9, or 10) where NCAA coaches will look at you seriously as a potential recruit. That could change with excellent coaching, ample practice time (a minimum of 10-15 hours per week), and good training facilities - which would include pits.
All that said - competition for a Men's NCAA scholarship is HIGHLY competitive nowadays. Most men's programs have 4-6 full scholarships - and those are often sub-divided. With 16 teams that a max of around 96 scholarships... and only around a quarter of those (about 24) open up each year as seniors graduate. So - getting into the top 24 range nationally is tough!!! NOT impossible(!) - I've seen it happen.
Having a strong high school academic background is going to be important as well... you'll be competing with the cream of the crop in both athletic and academic areas.
On the "cha-ching" (Gym Expense) bit. Don't be discouraged. I was in a similar situation when I was in high school. I started at age 15 also BTW - so I do understand your concerns. It was years (decades!) ago... but I taught kids recreational classes at my gym five days a week, helped set-up and tear down the gym every weekend (it was a YMCA program) and that was a trade-off for my training, meet entry and team travel fees. Find a great gym with a good coach. Ask about "scholarship" possibilities at that gym - trade instruction/gym maintenance (whatever!) so that you might have a chance at making your dream come true. The cheerleading/diving thing - that might serve as a fall-back tho those scholarships are competitive as well.
On the request for level assessment: USAG Levels this skill is used in)
FLOOR:
Cartwheel; (4 & 5)
back walkover
front walkover
handstand (Step to handstand... roll; 4 & 5 - - 360° pirouette; 5 & 6
back handspring
round- off back handspring; Bonus in 4, required in 5, 6, & 7
round-off double back handspring Bonus in 5
front handspring; required in 5, 6 & 7
back tuck; required in 6 & 7
POMMEL HORSE:
around the worlds??? Are these double leg circles - I don't think so....
pike hold (L-sit?)
tuck hold
pendular swings to leg cut forward and backward - I call it cut-swing-cut you said "those ones where you lift one arm and kick a leg over and back and then do it on the other side"
All these are progressions for level 4 and above
RINGS:
Straddle hold??? (don't know what this is...)
Pike hold (L-sit?)
pike to inverted (hang?)
Tuck hold
Back roll (backward drop to inverted hang from support? - or skin the cat (to German hang)?)
inverted hang?
VAULT:
Front handspring; 6
round-off; 7
PARALLEL BARS:
straddle swings
pike hold (L-sit?); 4 and up
tuck hold?
shoulder stand
shoulder stand to roll
shoulder hops? All the above are progressions for levels 4 and up
HIGH BAR:
Pullover; 4, 5 & 6
back hip circle; 4 & 5
tap swings are getting high almost for them giants; 4-6
close to kip; 5 -7
front hip circle (not used much in men's high bar)
muscle ups; this is a Stills Rings skill - level 5
Send me your email address (mine is sbonham at georgiasouthern dot edu) and I'll send you a complimentary Ganba Gymnastics Illustrated Guide (Sequence Illustrations of USAG Men's Compulsory Routines Levels 4-7). It includes the names of the skills and should be helpful.
All the Best,
Steve
sbonham@ georgiasouthern.edu