My suggestion is to use half of your savings ($50k) for your down payment and leave the rest for your emergency fund, without any investment. This assuming your comment about the $100k actually being your savings is true.
First off, never assume you are going to be constantly employed for any length of time. Not 15 years, not 15 months. You likely don't know your employer that well, you don't know what the economy will do, you don't know if someone is going to or is already embezzling from your company, you could become disabled in a car wreck, there's just too many reasons why your job could be at stake to blindly assume that you're going to be steadily employed.
Anything you put down on a house is more money you save. Only buy a house 100% financed if you absolutely have to. It's great it's an option, but it sounds like you don't need to use that option. Also, interest rates may be at 3-4% right now, but that's entirely dependent on your credit history. In the USA at least, the only people who get the advertised rates are those with perfect or nearly perfect credit ratings. Everyone else gets a higher interest. If you are one of those people with perfect credit, great, but you still want to save money on interest payments for the life of your loan.
Using the mortgage calculator above, at $300k, 3% interest rate, putting $50k down saves you around $44k in interest payments. Putting all $100k down saves you around $110k in interest, but you are left open to financial problems by not having an emergency fund until you can replenish it. I don't recommend living without an emergency fund. There's too many things that can cause you misery without one.
FYI, I'm not affiliated with NerdWallet, they just had the options in the calculator I wanted to use, such as HOA, insurance, and property taxes.
As far as investing in stocks, I consider this the same as gambling: Only spend what you can afford to lose. Since you don't say how much you earn, which you don't have to, I'm going to assume you aren't super rich, so you need the money you have. Yes, people get rich off stocks, but usually only after a lot of mistakes and also by putting a lot of time and money into the stock market.
If you really feel the need, put maybe $2000 into a trading account and see how you do. If you do well, keep rolling that money into more trades. If you do really well, you might consider adding another $10k into the account and maintain your method of good trades. Too many people go on "autopilot" after winning some good trades and think that they are all of a sudden able to win at anything, while putting a lot more down, only to lose it all. Just, be careful.
There's 1 million other versions of "how best to proceed", and even 1000 variations on my suggestion. It's really up to what you are comfortable with. There's a lot of risk with the stock market, much less with your mortgage, and not much risk with your savings. What you're really asking is "where do I put my money that the risk pays off", and only you can decide what risk you are actually comfortable with.