Don't worry about broadcasting your transaction.
First, you should know whether you have any bitcoin or not. If you really have bitcoin and you own the private key, you will be able to spend the fund.


From "blockheight": 4111
As I said above, the data you shared doesn't match the blockchain.


What's the name of the software?

You are looking at a wrong transaction.
In the transaction you shared, 17.61386661 BTC has been sent from a segwit address.

According to the data shared by OP, the 50 BTC have been mined in the block number 4111. Click here to see the coinbase transaction of that block.
The 50 BTC mined in that block is still unspent.

I don't really understand what you mean by "it takes 10 years to update".
Are you trying to download the blockchain in bitcoin core and it says it takes 10 years to be synced? If so, it shouldn't take that much.


No.
Either you don't have bitcoin at all or you are looking at a wrong address.

According to the data you shared, the 50 BTC have been mined in the block number 4111.
Bitcoin generated on that block haven't been moved at all and are still in the address 1C7JtZUBeSPy7eZQ9vuCcaHRzYjKmnXACJ. This address doesn't match what you shared above.
The transaction ID you shared above is also different from the hash of coinbase transaction of that block.

Where did you take this data from?

If you can't store 400 GB of data, you can enable pruning in bitcoin core. In this way, you don't have to store that large data. Note that even if you run a prune node, you still have to download the full blockchain.

The other only solution is to export your private key(s) from Armory and export it into a SPV wallet like electrum.
In SPV wallets, you don't have to download the blockchain.

Warning:
Download electrum only from its official website and do not forget to verify your download.

The post has been edited.
Sorry. As mentioned below by BitMaxz, Armory doesn't work if you download the blockchain in prune mode.

Note that for signing a message from a bitcoin address, you need to have full control over the address. This means that you must own the private key of the address.
The deposit address an exchange gives you isn't owned by you. That's owned by the exchange and you can't sign a message from that.

Can you sign a message from any of addresses you have used before?
Staking a new address without proving your ownership over the address you have staked before defeats the purpose of staking a bitcoin address.

You can use ninjastic.space to see the addresses you have used before.

Have you installed bitcoin core?
To see the receiving transaction on Armory, you have to install bitcoin core on the same device and download the full blockchain.

To make sure you have received the fund, you can search for your receiving address on a block explorer like blockchair.com and check the balance and transactions history there.

This is wrong.
Even if the winner has purchased all the tickets (which doesn't seem to be the case), he/she has spent only 1.2175 BTC for buying them, not over 10 BTC. Each golden ticket costs 25,000 satoshi.
4870*0.00025 BTC = 1.2175 BTC

If the winner has earned all the tickets through wagering, he/she wagered 24.35 BTC for earning those tickets. You get 1 golden ticket for every 0.005 BTC you wager.
4870*0.005 BTC = 24.35 BTC

It shouldn't be an edited image. The user has posted the winning seeds as well.
Anyways, even if that's an edited image and the user has changed the seeds to cheat freebitco, the game is still provably fair.

There is no doubt that you always have the chance of rolling the number 10000 and winning the biggest prize.
If you have any doubt on fairness of the game, I recommend you to click on "This game is provably fair" button and read the details on how the rolls number are calculated.

---

Edit:
I just noticed that the winner of round 7 of the Lamborghini lottery has been determined.  
Someone with 4,870 tickets has won the lottery.

Below is the announcement they have made about zero trading fee for 13 bitcoin pairs. It does't say anything about holding BNB and that doesn't seem to be required.
Binance Launches Zero-Fee Bitcoin Trading

I stopped using binance after compulsory KYC and will never use it. So, I can't confirm this myself.

The answer to this question should be given by TheQuin.
I can only say that there's nothing to worry about. The lottery is provably fair and there is no way to cheat the users. The only problem is that participants have to wait some more time.

Whenever they publish the list, the hash of the block that will be used for the calculations is unknown.

A new bitcoin block is mined every 10 mintues on average. So, mining 12 blocks should take only about 2 hours and that's not what has caused the delay.
Note that tickets number list hasn't been published yet. They should publish the list in their twitter and 12 bitcoin blocks should be mined after the list published.

Musk caused only some short-term manipulations and can no longer have any significant effect on the market.
Also, mining is still profitable. As mentioned by o_e_l_e_o above your post, the total hash power is now at its all time high. That couldn't happen, if mining wasn't profitable.
The following chart shows bitcoin total hash power changes over time.


blockchain.com

Depending on how the exchange works, it may be possible.
After you make a deposit to your exchange deposit address, the exchange sends the fund to their own wallet.
If the wallet used by the exchange is publicly known, it should be easy to find out what exchange the address belongs to.

Use walletexplorer.com.

The winner is determined through a provably fair lottery. But that doesn't mean it's done completely automatically.
The first step is that they will publish a list including user ID of all the participants and their lottery tickets in their twitter. That list hasn't been published yet.
After the list is published, you should wait about two hours for 12 bitcoin blocks to be solved. After that the winner will be determined through mathematical calculations.

The address has received 944 BTC and has spent all of that.

Note that when you withdraw from an exchange, the withdrawal isn't made from the address you deposited to.
When you deposit fund to your deposit address, exchange sends the fund from your deposit address to their own wallet and when you request a withdrawal, the exchange send you the fund from their own wallet.


Exchanging bitcoin to usdt or exchanging any other coin/token to any other coin/token is done on the exchange, not on the blockchain. So, trades are not recorded in the blockchain at all.

Here we are talking about bitcoin and note that there is no connection between bitcoin and ethereum blokchains.

If you send 1 BTC to a bitcoin address (whether it's owned by an exchange and you want to trade with your bitcoin or it's owned by someone and you want to buy something), an output with the value of 1 BTC is created and will be owned by the owner of that address. If you search for the receiving address in a blockchain explorer, you will see that address has received 1 BTC. Note that your transaction may include an extra output as the change.
It's also possible to make a transaction with multiple outputs. For example, you can pay 1 BTC to address A, 2 BTC in address B and 3 BTC to address C in a single transaction.

This is correct.
That address received 8 BTC in e3a1....822c and spent that in 5c14....a285.


You are right that that address has spent 8 BTC in that transaction, but that doesn't mean there must be an output with the value of 8 BTC as well.
A bitcoin transaction can have multiple inputs and multiple outputs and each of them can have different values.

Note that any bitcoin transaction can have multiple inputs and multiple outputs.
There are 62 inputs and 57 outputs in the transaction in question and one the inputs is the 8 BTC you are referring to.


That's because there's no address receiving 8 BTC at all.
It's not that there should be a 8 BTC ouput, because there's a 8 BTC input.

I don't think there's any syncing issue.
OP said that he can't find the address he is looking for at all. He also said that he can see the balance on the watch-only wallet.
Most probably, OP is using a seed phrase which belongs to another wallet.