Reader Case Study: Invest more or save liquid--on one income?

[PlanB]

I've been reading MMM for a year now, and I'd love more investment/savings information in regards to our current situation.
I (age 32) work full-time and my husband (age 34) stays at home with our children (ages 5, and almost 2), he also bar tends 2 shifts a week.
The details are as follows:
Income:
Myself: 3,000 take home
Husband: $800 take home (varies)
Total: $3,800

Current monthly expenses:

House mortgage (includes taxes/insurance): $1500
Utilities (gas/elec/water): $150
Phone: $55 (for two, just switched to Ting from Sprint)
Netflix: $7
Internet: $50
Car insurance: $250 (paid in full every 6 months)- paid with extra bi-weekly pay, not included in income.
Heating: $1500 (for 2014 winter on old furnace, which we will change at some point in future to gas furnace)- paid with extra bi-weekly pay, not included in income.

Variable:
Groceries: $500 (includes organic veg/meat, and alcohol) --this will decrease in summer due to garden. We have an herb/tomato/salad garden at home and a free plot in a community garden in town, where we grow our larger veg). Working on canning more this year.
Gas: $200 (2 cars; I just started biking to work, this should decrease a bit)
Misc: $200 (includes medication, riding/barn fee, etc.)
We put all expenses (that we are able) on a CC and pay off fully each month. Rewards go to help pay for Christmas/gift expenses. I create many gifts, to hold down costs.

Total: $2662
Savings: $1138

Assets:
401k: $35,000 (currently contributing 6% with 3% match)
1999 Range Rover: fully paid. Worth: 6,000
2000 Saab 9-3: fully paid. Worth: 4,000
House: value: 310,000 (purchased for 289k, with 58k down)
Savings: 10,000 (should I move some of this automatically to invest?)
Investing: 10,000 in VFINX-- is this the right place to be?
Bonds: 4,000 in EE bonds, at various maturation rates from 2018-2023. Earning 4% flat. Move these? Where to?

Liabilities:
Mortgage on house: 225,000 (30 year 3.2%, 29 years left)

My question is: Where should we best allocate savings? We do have some home improvement projects (insulation, flooring, conversion to gas furnace, etc.) that we'd like to do in the next 1-10 years, but nothing is a huge rush (insulation excepted), and we DIY what we are able to keep costs down. We currently have a 10k emergency/liquid fund. Should I start an auto-invest with our monthly savings? Is the Vanguard index fund a good one to put in and forget about?
Now to throw a (great) wrench in the works, what would you do differently if a 100k gift came into play? This will happen, the timing is the variable. I'd say it's within the next 10 years. Would you invest? Throw on mortgage?

*I'm not specifically looking to RE, but would like the option and so have been working to optimize our income/spending/investments. Our income might increase when both kids are in school with my husband working more, and my position will definitely increase in the next 3-5 years. I feel like we're doing okay with spending, but am open to all suggestions.
**Sidenote: Our mortgage may seem large, but is actually a very inexpensive housing situation in our HCOL area (avg house price 500k). We decided to pay more up front (and bought a lovely house ("needing" only superficial renovations) in order to be very close to work/family/friends. This is the one thing that won't change, for now.
Thanks in advance for any/all advice and suggestions!

[NewStachian]

Question 1: I think this is really 2 parts: what funds should you buy, but also what vehicle should you use?

Funds: I've heard a lot of people use that fund and it seems like they like it. I use the IVV ETF as my bread and butter. It's iShares S&P 500 ETF (Which I think is almost the same as the one you mentioned, but not sure). Its ratio is 0.07%. Large cap is a good place to start when getting into investing and only want to buy 1 fund/ETF. (But I'll let some more people with strong opinions weigh in on that)

Vehicle: I recommend putting enough in to get your company's 401(k) match (free money), then max out your ROTH IRA's (11k for the year). Since you are in a lower tax bracket, post-tax money compounding over time will most likely be the best bet for you. Plus, you can take principal out any time down the road. While, it's best to keep this money for retirement, it is much more flexible to tap into later. Once you build up a ROTH IRA, you can also get more aggressive and keep less cash on hand since the ROTH is a solid safety net.

Question 1B: I highly recommend you automate your monthly investing so it goes automatically in the fund that you select. This has a ton of benefits, but I won't list them all here.

Question 2: 100k gift: BLOWOUT IN VEGAS! (kidding) I am in a very similar situation to you: 29 years left with a low interest rate. Many Mustachians on this forum will tell you to get rid of your mortgage. I personally believe that a low interest mortgage is good. If you assume historical inflation and market return, then it will be financially optimal to pay minimum on the mortgage and, instead, put that extra money into the market. Having secured debt is also a great hedge against inflation. But, some people hate having a lot of debt and it takes a toll on them emotionally. So, it's really up to you. If you put 100k into your mortgage instead of investments it's certainly not bad... just potentially sub-optimal financially over the long run. I might pay the mortgage off in full in 15-20 years. By that point I have capitalized on all that compounding from today's investments and it will be a trivial mortgage to pay off.

Warning: make sure you know the tax situation with the 100k gift before putting that money to work.

[PlanB]

Thanks for the reply, NewStachian! I wasn't even thinking of Roth IRA's for some reason... so thanks for putting that on my radar. I will look into funding one for 2014 now.
I'm looking forward to hearing about auto-investing and the best vehicle for it.

Thanks!