To answer the questions you have. All this is from my experience. You will find very conflicting ideas on forums and in your research. Learn good tank husbandry and the rest is pretty easy
IMO.
1. Filter- what's the best to use on a 10g tank? Not diving into sumps..
I have used a HOB (hang on back) marineland biowheel filter rated for 20 gallons with good success. Some say they will produce lots of nitrates. Hasn't been my experience, as long as you clean the filter of gunk that does build up over time. Toothbrush is your friend 
2. I have a heater for the 10g tank. Need to check if it is also applicable to saltwater.
Is more than likely fine for saltwater. I would recommend one that has adjustable temp and about 50w higher as keeping the temp stable is easier with a heater that is a little more powerful. AKA if you have a 10g, 50 watt, consider 100 watt.
3. Powerhead- do I need one?
Depends. I had a filter for a 20 gallon in my 10 gallon and that created enough water movement. Wouldn't hurt to get a small one for added movement if your filter isn't giving you enough to your liking. 1 nano power head should be all you need if you get one.
4. Mini skimmer? Do I need one for a tank this small? If it helps with filtration I'm willing to invest.
No you don't need one on a tank this small. While skimmers do a great job at what they do, you will achieve just as good results with regular water changes. Now if you had a 30g or higher I would say yes.
5. Probably the most important- how do you Fishless cycle a saltwater tank? All you salty gurus probably know the best ways to do it!
I drop a piece of raw shrimp in the tank and let it sit for a week then take it out. You can also dose ammonia to get it started as well, but the shrimp works just as well.
6. Doing white sand. It's leftover in the bag from a freshwater tank. Will this work or do I need a special sand?
Thats a big debate. I bought a 20lb bag of live sand from Petco when I setup mine. It was like $22. I am a fan of live sand. Some are not. Personally your main bio filtration will come from your rock, not so much your sand. But for a small tank, a 1 time purchase of $22 to get going isn't much.
7. Live rock or dry rock? I'm clueless. My
LFS sells live rock at about $8.99 a pound. Sounds a bit high but it's my only choice.
I use dry rock. Main reason is you never know what you get with live rock. Dry rock only gets what you put in your tank, while live rock has anything that is on it, whether you want it or not. It will take time for your dry rock to become "live" though. Its a personal opinion and choice, but either is fine. 1/2lb per gallon is what I would put in. So about 5-7lbs. Remember rock displaces water so the more rock, the less water volume.
8. Lighting! No idea where to even start... Don't plan on corals till I get some major experience. Unlike the $5 freshwater plants I can test run, corals cost money :p -
For starting out, I would get a small LED light. You can get LED lighted 10g hoods for not very much or just the light if you already have a hood. Once you get to a place where you want to do more, like corals, you can look into t5 or LEDs good for reefs.
9. I'm only think like a goby and shrimp. Any suggestions welcome but I want to start cycling before we start talking critters. -
There are plenty of Gobys that would great for a 10 gallon. Just remember to have a hood and try to cover all open gaps. I had a diamond goby that supermaned out of my tank because I didn't have a small gap covered.
Water changes. This will be the most important part of your tank, mainly because it is a smaller tank. With rock and sand you will have around 8 gallons of water in there. Water parameters can get out of hand quick if this step is not done regularly. I would suggest a 20% water change weekly, or 50% every other. 2 gallons weekly, or 4 every other. And if you want a 50% weekly would be even better. You have two options. 1. Make it yourself, or 2. buy it from your
LFS. It is easy to make, but you need to use
RODI water. You can get a 50lb bag mix of Instant Ocean salt mix (purple bag) for around $20. 1/2 cup per gallon will give you a salinity of 1.022 to 1.023. Directions are on the bag.
It is easier to buy the water. Just is. Its already pre mixed up, you just heat it up to your tanks temp and go. My
LFS charges $1 a gallon.
It will save you money to buy
RODI water at your
LFS (mine charges .25 cents a gallon) and mix it yourself. Just to give the math (Buying 50 gallons of saltwater at $1/gallon would be $50. Buying 50 gallons of
RODI water would be 12.50+~$20 for the mix comes out to be ~$32.50. Just something to consider.
Not overfeeding, and regular water changes, and cleaning your filter (replacing cartridges too) and you shouldn't have a problem. Some will say 10 gallons are hard to keep. I didn't think so, as I started many years ago with a 10 gallon. Good tank husbandry and you will be fine.
could I keep a cleaner and a sexy together?
I would stick with what you have and make the one time purchase of some reef capable LEDs when you do decide to get coral. They are cheap on eBay. I agree with the weekly water change being your best friend. It's easy to mix yourself. Oh, and I'm not sure about the sexy shrimp and cleaner shrimp compatibility. Just remember that sexy shrimp can become a meal to some inhabitants.
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