I see lots of good advice above. Here is my take on your issues.
The 20 amp outlet in your bathroom is protected by a 20 amp circuit breaker in your home's service entrance. Breaker will trip if you overload it. One or two trips will not hurt anything. Tripping says you are trying to use more than the outlet is capable of.
Using the 30 amp RV cord with a 15/20 amp adapter will work well. If you use a common 120 volt extension cord, make sure it is at least 12 gauge. 10 gauge will be more efficient.
Most extension cords are 14 gauge lighting cords. 14 gauge is too small. Cords that do not list the gauge size on the packaging are usually 14 gauge. Find one the specifically says 10 gauge.
The 30 amp service in your Forest River TT also has circuit breakers. They are not shown in your pictures. The circuit breakers will protect your TT circuits.
2003 forest river 23'. I attached pictures of the amps I have inside my trailer. I bought a 30ft 30amp extension wire to connect power to my house. My questions are;
1) is the 30amp power cord that I bought would be good to use for my trailer?
Most likely the answer is "yes". A picture of the plug and socket on the cord would confirm. 30 amps is the correct current.
The Forest River probably came with a 30 amp cord that attaches to a plug on the side of the TT. The other end has a three blade "RV" 30 amp 120 volt plug. It will plug into some generators or a grid shore power pedestal.
You can have a socket wired into your house that will supply 30 amps at 120 volts. As posted above, be careful when having an electrician install the socket. Not all qualified electricians are aware the RV needs 120 volts. They are usually more familiar with 240 volt 30 amp outlets and have made expensive mistakes for some people.
2) for now, i am plugging it to a bathroom outlet which I assume it has 20amps, will this be okay? Can I run my AC with it? What else can I use it with together?
Yes it is OK to plug into your bathroom outlet using an adapter on the end of the 30 amp cord. The home circuit breaker will trip if you try to draw too much current.
Your A/C will probably work OK plugged into that outlet. The A/C will draw about 10 amps when running.
I see conflicting advice posted above. Be aware the home circuit breaker will trip if you exceed 20 amps. So any other heating or cooling appliances like a hair dryer run at the same time on the same circuit will probably trip the home breaker.
The distance from your home electrical service entrance to your TT matters. Wiring code specifies 12 gauge wire for up to 50 feet. Beyond that, 10 gauges wire is required. The 30 amp RV power cord is usually 25 feet and is 10 gauge wire. You can add another RV extension cord for another 25 feet and usually be under 100 feet. Further than that and you must consider other requirements.
3) on the picture, it says 45amps, does it mean that my trailer is running more than 30amps?
As posted above the picture shows the 12 volt converter/charger. It is a 45 amp 12 volt DC unit. It is great for charging a one or two battery, battery bank (70 to 200 amp hours total). It will fully charge a depleted battery bank in 14 to 18 hours. It can charge from 10% SOC (state of charge) to 80% SOC in about 4 hours.
Do the 18 hour full charge before storing the TT regardless of starting SOC. Storing fully charged is required for long battery life.
The converter/charger will also power 12 volt lights, water pump, furnace, vent fan, and control boards for furnace, water heater, and refrigerator.
4)I am planning to buy that generator on the picture, will that be enough to run all of my appliances in the trailer? (Microwave, refrigerator, 13000btu ac, lights, fans and wall outlets)
The generator in the picture will be fully adequate for your TT. Like your home, running absolutely everything at once will overload and cause the TT or generator circuit breaker to trip. However, people rarely do that.
Many campers run on battery and propane for days. I get 5 days using a 200 amp hour battery bank. I use propane for the refrigerator and water heater. Running the air conditioner or microwave requires the generator or plugging into shore power at the campground.
I have a generator. I camp in Wisconsin often. I have not used the generator except to test it for the last 15 years.
5) should I install a 30, 40, or 50am outlet in my house?
I am lost with these electrical stuff. I don't want to break any of my appliances. Please advice. Thanks for all your help!
Are you going to use the TT to live in? If so you would do well to install a 30 amp 120 volt outlet.
If you only need to maintain the battery and prepare the TT for travel, an existing 20 amp outlet should be fine.
If the TT is more than 50 feet from the house you may wish to have an electrical pedestal installed near the place you plan to store it.
Ask your electrician the difference in cost for a 20 amp or 30 amp outlet. You need 20 amps. 30 amps is more convenient.
I wish you good luck and happy trails ahead!